Tiger Algebra/core — Swahili @ Weblate: In geometry, a circle is a shape made up of …

Translation

Key CircleContent

English

In geometry, a circle is a shape made up of all the points on a plane at a fixed distance around a given point (the center). The equation for a circle is <math>(x-h)^2+(y-k)^2=r^2</math>, in which <math>h</math> and <math>k</math> represent the circle's center and <math>r</math> represents the circle's radius, the distance from the circle's center to any point on its perimeter. A circle with a center at <math>(4,5)</math> and a radius of <math>10</math>, for example, could be expressed as <math>(x-4)^2+(y-5)^2=100</math>
 
<img src="/images/circleImg1.png" alt="circle graph">
 
<img src="/images/circleImg2.png" alt="circle related terms diameter, radius, chord, secant, tangent">
 
Relevant terms: 
<ul>
<li>Center: The point around which a circle is formed. All points on a circle's perimeter are the same distance from the circle's center.</li> 
<li>Circumference: The distance around a circle.</li> 
<li>Radius: A line segment that lies between the center of a circle and any point on the circle's perimeter</li>. 
<li>Diameter: A line segment that lies between two points on a circle's perimeter and runs through the circle's center. It is equal to two times the circle's radius.</li> 
<li>Chord: A line segment that lies between two points on a circle's perimeter and does not run through the circle's center.</li> 
<li>Secant: A line that intersects two points on a circle's perimeter.</li> 
<li>Tangent: A line that intersects one point on a circle's perimeter.</li>
</ul>

Swahili

Katika jiometri, mduara ni muundo uliotengeneza na idadi zote kwenye nafasi kwa umbali wa kudumu karibu na nafasi fulani (kitovu). Equation ya mduara ni <math>(x-h)^2+(y-k)^2=r^2</math>, ambako <math>h</math> na <math>k</math> zinawakilisha kitovu chake na <math>r</math> inawakilisha radiusi ya mduara, umbali kutoka kitovu cha mduara kwa nafasi yoyote kwenye mipaka yake. Mduara na kitovu kwenye <math>(4,5)</math> na radiusi ya <math>10</math>, kwa mfano, inaweza kuelezea kama <math>(x-4)^2+(y-5)^2=100</math> 
<img src="/images/circleImg1.png" alt="mchoro wa mduara">
 
<img src="/images/circleImg2.png" alt="maneno yanayohusiana na mduara diameteri, radiusi, chord, secant, tangent">
 
Maneno yanayohusiana: 
<ul>
<li>Kitovu: Nafasi ambayo mduara unajenga. Idadi zote kwenye mipaka ya mduara ziko umbali sawa kutoka kitovu cha mduara.</li> 
<li>Mzunguko: Umbali kuzunguka mduara.</li> 
<li>Radiusi: Sehemu ya mstari inayolala kati ya kitovu cha mduara na nafasi yoyote kwenye mipaka yake.</li> 
<li>Diameteri: Sehemu ya mstari inayolala kati ya nafasi mbili kwenye mipaka yake na inapitia kitovu chake. Ni sawa na ukubwa wa maredio mara mbili.</li> 
<li>Chord: Sehemu ya mstari inayolala kati ya nafasi mbili kwenye mipaka ya mduara na haipiti katikati ya mduara.</li> 
<li>Secant: Line inayokatisha nafasi mbili kwenye mipaka ya mduara.</li> 
<li>Tangent: Mstari unaokatisha nafasi moja kwenye mipaka ya mduara.</li>
</ul>

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Key	English	Swahili
AbsoluteValueEquations	Absolute value equations	Usawa wa thamani kamili
AbsoluteValueInequalities	Absolute value inequalities	Utofauti wa thamani kamili
AbsoluteValueEquationsContent	<h3>Absolute value</h3> Absolute value (sometimes called modulus or magnitude) is how far a number, term, polynomial, or expression is from zero, regardless of whether it is positive or negative. For example: <math>4</math> and <math>-4</math> are the same distance from <math>0</math>, so they both have an absolute value of <math>4</math>.<br><br> <img width="500" alt="Absolute value" src="https://user-images.githubusercontent.com/56831176/146247143-c8db661d-83ab-457f-9087-e9bc5cfa4ba3.png"><br> Absolute value is represented by two bars, one of each side of the number, term, polynomial, or expression. For example, the absolute value of <math>-4</math> would be written as <math>\|-4\|</math><br><br> <h3>Properties of absolute values</h3> <ul> <li>Non-negativity: <math>\|x\|≥0</math><br> Absolute value is always non-negative, meaning it <b>always</b> yields zero or a positive.</li><br> <li><math>\|x\|=\sqrt{x^2}</math>: Squaring a number makes it positive (or zero if the number is zero), and by taking the square root of a squared number we get a positive solution (or zero if the number is zero). This only works when <math>x</math> is a real number.</li><br> <li>Multiplicativity: <math>\|x\cdot y\|=\|x\|\cdot\|y\|</math><br> The absolute value of a product of two numbers equals the product of the absolute value of each number.</li><br> <li>Subadditivity: <math>\|x+y\|≤\|x\|+\|y\|</math><br> The absolute value of the sum of two real numbers is less than or equal to the sum of the absolute values of the two numbers.</li><br> <li><math>\|x\|=y → x=\pm y</math> or <math>\|x\|=\pm x</math>: If the absolute value of <math>x</math> equals <math>y</math> then <math>x</math> equals plus or minus <math>y</math>. This rule is used for solving most absolute value questions.</li> </ul> <br><br> <h3>Absolute value equations</h3> Absolute value equations are equations in which the variable is within an absolute value operator.<br> For example: <math>\|x-4\|=10</math><br> Because the value of <math>x-4</math> can be <math>10</math> or <math>-10</math>, both of which have an absolute value of <math>10</math>, we need to consider both cases: <math>x-4=10</math> and <math>x-4=-10</math>. This can also be written as <math>x-4=\pm10</math>.<br><br> So, <math>\|x-4\|=10</math> has two solutions:<br> <math>x-4=10</math> → <math>x=14</math><br> <math>x-4=-10</math> → <math>x=-6</math><br><br> Because absolute values are always non-negative, it is possible to have equations with no solutions.<br> For example: <math>\|x - 5\|=-9</math><br><br> Absolute Value Equations and Inequalities are solved and explained step by step by the Tiger Algebra Absolute Value module.	<h3>Thamani kamili</h3> Thamani kamili (wakati mwingine inaitwa modulus au ukubwa) ni kwa umbali gani namba, istilahi, polynomial, au kielelezo cha hesabu kinapatikana kutoka katika sufuri, haijalishi ikiwa ni chanya au hasi. Kwa mfano: <math>4</math> na <math>-4</math> zote zina umbali sawa kutoka <math>0</math>, hivyo zote zina thamani kamili ya <math>4</math>.<br><br> <img width="500" alt="Absolute value" src="https://user-images.githubusercontent.com/56831176/146247143-c8db661d-83ab-457f-9087-e9bc5cfa4ba3.png"><br> Thamani kamili inawakilishwa na mistari miwili, moja kila upande wa namba, istilahi, polynomial, au kielelezo. Kwa mfano, thamani kamili ya <math>-4</math> ingeandikwa kama <math>\|-4\|</math><br><br> <h3>Mali za thamani kamili</h3> <ul> <li>Nyeginezo: <math>\|x\|≥0</math><br> Thamani kamili siku zote ni nyeginezo, yaani hutoa sufuri au namba chanya.</li><br> <li><math>\|x\|=\sqrt{x^2}</math>: Kusawazisha namba hufanya iwe chanya (au sufuri kama namba ni sufuri), na kuchukua kipeuo cha namba ya kusawazishwa tunapata suluhisho chanya (au sufuri kama namba ni sufuri). Hii inafanya kazi tu hata kama <math>x</math> ni namba halisi.</li><br> <li>Multiplicativity: <math>\|x\cdot y\|=\|x\|\cdot\|y\|</math><br> Thamani kamili ya uzaji wa namba mbili sawa na uzaji wa thamani kamili ya kila namba.</li><br> <li>Subadditivity: <math>\|x+y\|≤\|x\|+\|y\|</math><br> Thamani kamili ya jumla ya namba mbili halisi ni ndogo au sawa na jumla ya thamani kamili ya namba mbili.</li><br> <li><math>\|x\|=y → x=\pm y</math> or <math>\|x\|=\pm x</math>: Kama thamani kamili ya <math>x</math> ni sawa na <math>y</math> basi <math>x</math> ni sawa na plus au minus <math>y</math>. Kanuni hii hutumiwa kutatua masuala mengi ya thamani kamili.</li> </ul> <br><br> <h3>Usawa wa thamani kamili </h3> Usawa wa thamani kamili ni usawa ambao kipengee kilicho na wazi kiko ndani ya kiendeshi cha thamani kamili.<br> Kwa mfano: <math>\|x-4\|=10</math><br> Kwa sababu thamani ya <math>x-4</math> inaweza kuwa <math>10</math> au <math>-10</math>, zote ambazo zina thamani kamili ya <math>10</math>, tunapaswa kuzingatia kesi zote mbili: <math>x-4=10</math> na <math>x-4=-10</math>. Hii pia inaweza kuandikwa kama <math>x-4=\pm10</math>.<br><br> Kwa hivyo, <math>\|x-4\|=10</math> ina suluhisho lisilo na kipengee:<br> <math>x-4=10</math> → <math>x=14</math><br> <math>x-4=-10</math> → <math>x=-6</math><br><br> Kwa sababu thamani kamili siku zote ni nyeginezo, inawezekana kuwa na usawa bila suluhisho.<br> Kwa mfano: <math>\|x - 5\|=-9</math><br><br> Usawa wa Thamani Kamili na Utofauti watatuliwa na kuelezwa hatua kwa hatua na moduli ya Thamani Kamili ya Tiger Algebra.
AddingSubtractingFindingLeastCommonMultiple	Adding subtracting finding least common multiple	Kuongeza kutoa kugundua kikatili cha chini
AddingSubtractingFindingLeastCommonMultipleContent	The Tiger Algebra Solver shows you how to add and subtract fractions and how to find their least common multiple for example:	Mfumo wa Algebra wa Tiger unaonyesha jinsi ya kuongeza na kutoa sehemu na jinsi ya kugundua kikatili cha chini kwa mfano:
ApproximationTopic	Approximation	Makadirio
Approximation	Approximation	Makadirio
ApproximationContent	Approximation in mathematics is defined as a quantity close in value to but not the same as a desired quantity. Two mathematical symbols denote approximate equality: a wavy equal sign (≈) and a dotted equal sign (≒ or ≓). <br><br> Approximation is often used when an precise form or integer is not known or easily obtainable as well as with irrational numbers, such as π. Approximation can turn a complex calculation, for example, involving decimals into one that uses integers, or whole numbers, making it easier to solve. This is achieved by rounding the decimal values before performing any further operations.	Makadirio katika hisabati ni kama kiasi cha karibu kwa thamani lakini si sawa na kiasi kinachotakiwa. Alama mbili za hisabati zinaonyesha karibu usawa: alama ya wavy ya usawa (≈) na alama ya nukta ya usawa (≒ au ≓). <br><br> Makadirio mara nyingi hutumika wakati fomu halisi au namba haijulikani au haipatikani kwa urahisi pamoja na nambari zisizo za kawaida, kama π. Makadirio yanaweza kugeuza hesabu ngumu, kwa mfano, ikijumuisha desimali kuwa moja ambayo hutumia integers, au nambari nzima, kuifanya iwe rahisi kutatua. Hii hutimizwa kwa kuzunguka maadili ya desimali kabla ya kufanya shughuli zozote za ziada.
ArithmeticSequence	Arithmetic sequences	Mfululizo wa hesabati
ArithmeticSequenceContent	An arithmetic sequence, or arithmetic progression, is a set of numbers in which the difference between consecutive terms (terms that come after one another) is constant. This difference is called the common difference. For example, all of the consecutive terms in the arithmetic sequence:<br> <b><math>1, 4, 7, 10, 13, 16, 19, . . .</math></b><br> share a common difference of <math>3</math>.<br> Note: The three dots (. . .) mean that this sequence is infinite.<br><br> Though others can also be used, the following variables are typically used to represent the terms of an arithmetic sequence:<br> <math>a_1</math> represents the first term of the sequence. In the example above, <math>a_1=1</math><br> <math>a_n</math> represents the nth term (a term we are trying to find).<br> <math>d</math> represents the common difference between consecutive terms. In the example above, <math>d=3</math><br> <math>n</math> represents the number of terms in the sequence. In the example above, <math>n=7</math><br><br> The standard form of arithmetic sequences can be expressed as: <math>a, a+d, a+2d, a+3d, a+4d, a+5d . . .</math><br> <math>a</math> represents the first term and is sometimes written as <math>a_1</math>.<br> <math>d</math> represents the common difference.<br><br> <b>Formulas</b><br><br> <b>Finding any term (<math>a_n</math>) in an arithmetic sequence:</b><br> <b><math>a_n=a+d(n-1)</math></b><br><br> <math>a\:</math> represents the first term.<br> <math>d\:</math> represents the common difference.<br> <math>n\:</math> represents the position of a term in the sequence.<br> A sequence with <math>\:n\:</math> number of terms would be written as:<br> <math>a, a+d(2-1), a+d(3-1), a+d(4-1), a+d(5-1), a+d(6-1) ... a+d(n-1)</math><br> in which the last term's common difference is multiplied by <math>\:n-1\:</math> (because <math>\:d\:</math> is not used in the 1st term).<br><br> Example: To find the next term in:<br> <math>1, 4, 7, 10, 13, 16, 19...</math><br> which would be the 8th term, we would plug the following into the general term formula <math>\:a_n= a+d(n-1)</math>:<br> <math>a\:</math> (first term) <math> =1</math><br> <math>d\:</math> (common difference) <math> =3</math><br> <math>n\:</math> (term number) <math> =8</math><br> This would give us:<br> <math>a_8=1+3(8-1)\:</math> which we could solve to get <math>\:a_8=22</math>.<br> So, our sequence would be: <math>\:1, 4, 7, 10, 13, 16, 19, 22...</math><br><br> <b>Finding the sum of all the terms in an arithmetic sequence:</b><br> <b><math>s=n(a_1+a_n)/2</math></b><br><br> <math>s\:</math> is the sum of the terms in the sequence.<br> <math>a\:</math> represents the first term.<br> <math>n\:</math> represents the position of a term in the sequence.<br> <math>d\:</math> represents the common difference.<br> Example: To find the sum of:<br> <math>1, 4, 7, 10, 13, 16, 19...\:</math> we plug the following into the sum formula <math>\:s=n(a_1+a_n)/2</math> :<br> <math>n\:</math> (total number of terms)<math> =7</math><br> <math>a\:</math> (first term)<math> =1</math><br> <math>a_n\:</math> (the last term)<math> =19</math><br> This would give us:<br> <math>s=7(1+19)/2\:</math> which we could solve to get <math>\:s=70</math>.<br> So, the sum of the sequence would be: <math>\:70</math><br> Tiger identifies arithmetic sequences and displays their terms, the sum of their terms, and their explicit and recursive forms.	Mfululizo wa hesabati, au maendeleo ya hesabati, ni seti ya nambari ambazo tofauti kati ya masharti yanayofuata (masharti yanayokuja baada ya moja nyingine) ni thabiti. Tofauti hii inaitwa tofauti ya kawaida. Kwa mfano, masharti yote katika mfululizo wa hesabati:<br> <b><math>1, 4, 7, 10, 13, 16, 19, . . .</math></b><br> wanashirikiana tofauti ya kawaida ya <math>3</math>.<br> Kumbuka: Dots tatu (. . .) zinaonyesha kuwa mfululizo huu ni usio na mwisho.<br><br> Ingawa wengine pia wanaweza kutumika, vigezo vifuatavyo kawaida hutumika kutambua masharti ya mfululizo wa hesabati:<br> <math>a_1</math> inawakilisha neno la kwanza la mfululizo. Katika mfano hapo juu, <math>a_1=1</math><br> <math>a_n</math> inawakilisha neno la nth (neno tunalojaribu kupata).<br> <math>d</math> inawakilisha tofauti ya kawaida kati ya masharti yanayofuata. Katika mfano hapo juu, <math>d=3</math><br> <math>n</math> inawakilisha idadi ya masharti katika mfululizo. Katika mfano hapo juu, <math>n=7</math><br><br> Fomu ya kawaida ya mfululizo wa hesabati inaweza kuonyeshwa kama: <math>a, a+d, a+2d, a+3d, a+4d, a+5d . . .</math><br> <math>a</math> inawakilisha neno la kwanza na wakati mwingine huandikwa kama <math>a_1</math>.<br> <math>d</math> inawakilisha tofauti ya kawaida.<br><br> <b>Fomula</b><br><br> <b>Kugundua neno lolote (<math>a_n</math>) katika mfululizo wa hesabati:</b><br> <b><math>a_n=a+d(n-1)</math></b><br><br> <math>a\:</math> inawakilisha neno la kwanza.<br> <math>d\:</math> inawakilisha tofauti ya kawaida.<br> <math>n\:</math> inawakilisha nafasi ya neno katika mfululizo.<br> Mfululizo wenye <math>\:n\:</math> idadi ya maneno ungeandikwa kama:<br> <math>a, a+d(2-1), a+d(3-1), a+d(4-1), a+d(5-1), a+d(6-1) ... a+d(n-1)</math><br> inayotoshea tofauti ya kawaida ya neno la mwisho inayozidisha <math>\:n-1\:</math> (kwa sababu <math>\:d\:</math> haitumiki katika neno la 1st).<br><br> Mfano: Kugundua neno linalofuata katika:<br> <math>1, 4, 7, 10, 13, 16, 19...</math><br> ambayo itakuwa neno la 8, tungechoma yafuatayo katika fomula ya neno kuu <math>\:a_n= a+d(n-1)</math>:<br> <math>a\:</math> (neno la kwanza) <math> =1</math><br> <math>d\:</math> (tofauti ya kawaida) <math> =3</math><br> <math>n\:</math> (namba ya neno) <math> =8</math><br> Hii itatupa:<br> <math>a_8=1+3(8-1)\:</math> ambayo tunaweza kutatua kupata <math>\:a_8=22</math>.<br> Kwa hivyo, mfululizo wetu utakuwa: <math>\:1, 4, 7, 10, 13, 16, 19, 22...</math><br><br> <b>Kupata jumla ya maneno yote katika mfululizo wa hesabati:</b><br> <b><math>s=n(a_1+a_n)/2</math></b><br><br> <math>s\:</math> ni jumla ya maneno katika mfululizo.<br> <math>a\:</math> inawakilisha neno la kwanza.<br> <math>n\:</math> inawakilisha nafasi ya neno katika mfululizo.<br> <math>d\:</math> inawakilisha tofauti ya kawaida.<br> Mfano: Kupata jumla ya:<br> <math>1, 4, 7, 10, 13, 16, 19...\:</math> tunachoma yafuatayo katika fomula ya jumla <math>\:s=n(a_1+a_n)/2</math> :<br> <math>n\:</math> (jumla ya maneno)<math> =7</math><br> <math>a\:</math> (neno la kwanza)<math> =1</math><br> <math>a_n\:</math> (neno la mwisho)<math> =19</math><br> Hii itatupa:<br> <math>s=7(1+19)/2\:</math> ambayo tunaweza kutatua kupata <math>\:s=70</math>.<br> Kwa hivyo, jumla ya mfululizo itakuwa: <math>\:70</math><br> Tiger inatambua mfululizo wa hesabati na kuonyesha maneno yao, jumla ya maneno yao, na fomu zao za wazi na za marudiano.
BasicComplexOperations	Basic complex operations	Operesheni za kawaida za tata
BasicComplexOperationsContent	A complex number is a number that can be expressed in the form <math>a + bi</math>, where a and b are real numbers and i is the imaginary unit, that satisfies the equation <math>x^2=−1</math>, that is, <math>i^2=−1</math>. In this expression, a is the real part and b is the imaginary part of the complex number. <br><br> The Tiger Algebra Calculator performs the basic operations on complex/imaginary numbers and shows you the step by step solution.	Namba tata ni namba inayoweza kuelezea kwa mfumo wa <math>a + bi</math>, ambapo a na b ni namba halisi na i ni kitengo cha kufikiria, kinachotimiza equation <math>x^2=−1</math>, yaani, <math>i^2=−1</math>. Katika expression hii, a ni sehemu halisi na b ni sehemu ya kufikiria ya idadi tata. <br><br> Kalkulatori ya Algebra ya Tiger inafanya operesheni za kawaida kwenye namba tata/zisizo halisi na inakuonyesha suluhisho hatua kwa hatua.
CancelingOut	Canceling out	Kufuta
CancelingOutContent	When simplifying fractions you can sometimes divide the top (NUMERATOR) and bottom (DENOMINATOR) of a fraction by the same number. This simplification of a fraction is referred to as canceling out/down . Common problems require you to write a fraction in its simplest terms. To achieve that, you must keep canceling the fraction until it cannot be cancelled anymore. <br><br> The Tiger Algebra Solver and Calculator can help you check your answers when canceling out.	Unapokuwa unaweka ulaini au ukwaju namba mara nyingine unaweza kugawa juu (Numereta) na chini (Denomineta) ya ulaini au ukwaji na idadi. Hii ulaini au ukwaju wa sehemu inajulikana kama kufuta chini. Matatizo ya kawaida yanakuomba uandike ulaini au ukwaji katika m terms yake ya kawaida. Kufanikisha hilo, lazima uendelee kufuta ulaini au ukwaji hadi isiweze tena kufutwa. <br><br> Angalia Toleo lako la Algebra na Calculator kusaidia unapobadilisha majibu.
CircleTopic	Circle from equation	Mduara kutoka kwenye equation
CircleContent	In geometry, a circle is a shape made up of all the points on a plane at a fixed distance around a given point (the center). The equation for a circle is <math>(x-h)^2+(y-k)^2=r^2</math>, in which <math>h</math> and <math>k</math> represent the circle's center and <math>r</math> represents the circle's radius, the distance from the circle's center to any point on its perimeter. A circle with a center at <math>(4,5)</math> and a radius of <math>10</math>, for example, could be expressed as <math>(x-4)^2+(y-5)^2=100</math> <br> <img src="/images/circleImg1.png" alt="circle graph"> <br> <img src="/images/circleImg2.png" alt="circle related terms diameter, radius, chord, secant, tangent"> <br> <b>Relevant terms:</b><br> <ul> <li><b>Center</b>: The point around which a circle is formed. All points on a circle's perimeter are the same distance from the circle's center.</li><br> <li><b>Circumference</b>: The distance around a circle.</li><br> <li><b>Radius</b>: A line segment that lies between the center of a circle and any point on the circle's perimeter</li>.<br> <li><b>Diameter</b>: A line segment that lies between two points on a circle's perimeter and runs through the circle's center. It is equal to two times the circle's radius.</li><br> <li><b>Chord</b>: A line segment that lies between two points on a circle's perimeter and does not run through the circle's center.</li><br> <li><b>Secant</b>: A line that intersects two points on a circle's perimeter.</li><br> <li><b>Tangent</b>: A line that intersects one point on a circle's perimeter.</li> </ul>	Katika jiometri, mduara ni muundo uliotengeneza na idadi zote kwenye nafasi kwa umbali wa kudumu karibu na nafasi fulani (kitovu). Equation ya mduara ni <math>(x-h)^2+(y-k)^2=r^2</math>, ambako <math>h</math> na <math>k</math> zinawakilisha kitovu chake na <math>r</math> inawakilisha radiusi ya mduara, umbali kutoka kitovu cha mduara kwa nafasi yoyote kwenye mipaka yake. Mduara na kitovu kwenye <math>(4,5)</math> na radiusi ya <math>10</math>, kwa mfano, inaweza kuelezea kama <math>(x-4)^2+(y-5)^2=100</math><br> <img src="/images/circleImg1.png" alt="mchoro wa mduara"> <br> <img src="/images/circleImg2.png" alt="maneno yanayohusiana na mduara diameteri, radiusi, chord, secant, tangent"> <br> <b>Maneno yanayohusiana:</b><br> <ul> <li><b>Kitovu</b>: Nafasi ambayo mduara unajenga. Idadi zote kwenye mipaka ya mduara ziko umbali sawa kutoka kitovu cha mduara.</li><br> <li><b>Mzunguko</b>: Umbali kuzunguka mduara.</li><br> <li><b>Radiusi</b>: Sehemu ya mstari inayolala kati ya kitovu cha mduara na nafasi yoyote kwenye mipaka yake.</li><br> <li><b>Diameteri</b>: Sehemu ya mstari inayolala kati ya nafasi mbili kwenye mipaka yake na inapitia kitovu chake. Ni sawa na ukubwa wa maredio mara mbili.</li><br> <li><b>Chord</b>: Sehemu ya mstari inayolala kati ya nafasi mbili kwenye mipaka ya mduara na haipiti katikati ya mduara.</li><br> <li><b>Secant</b>: Line inayokatisha nafasi mbili kwenye mipaka ya mduara.</li><br> <li><b>Tangent</b>: Mstari unaokatisha nafasi moja kwenye mipaka ya mduara.</li> </ul>
Combination	Combination	Uchanganyiko
CombinationTopic	Combination	Uchanganyiko
CombinationContent	In mathematics, a combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. In smaller cases it is possible to count the number of combinations. <br><br>Tiger Algebra calculates the number of combinations showing you the step by step solution. To activate, enter your input in one of the following forms: <div style="white-space: normal;"> <p>In combinatorics, a combination is a selection of items from a larger set, where the order of selection does not matter. Combinations are often used to count the number of ways to choose a subset of objects from a larger set.</p> <h2>Formula</h2> <p>The number of combinations of <math>k</math> items chosen from a set of <math>n</math> items (denoted as <math>C(n, k)</math> or <math>{n \choose k}</math>) is calculated using the combination formula:</p> <div style="margin-left: 20px;"> <math>C(n, k) = \frac{n!}{k!(n - k)!}</math>, </div> <p>where <math>n!</math> represents the factorial of <math>n</math>, defined as the product of all positive integers less than or equal to <math>n</math>.</p> <h2>Properties</h2> <ul> <li>The number of combinations is always a non-negative integer.</li> <li>Combinations are unordered, meaning that selecting the same set of items in a different order does not create a new combination.</li> <li>The number of combinations is often used in probability calculations and counting problems.</li> </ul> <h2>Example</h2> <p>Suppose we have a set of 5 letters: A, B, C, D, and E. We want to choose 3 letters from this set without regard to the order of selection. The number of possible combinations is:</p> <div style="margin-left: 20px;"> <math>C(5, 3) = \frac{5!}{3!(5 - 3)!} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10</math>. </div> <p>Therefore, there are 10 different combinations of 3 letters that can be chosen from the set {A, B, C, D, E}.</p> <p>Combinations are fundamental in combinatorial mathematics and have numerous applications in various fields, including statistics, computer science, and cryptography.</p> </div>	Katika hisabati, uchanganyiko ni njia ya kuchagua vitu kutoka kwenye mkusanyiko, ili kwamba (tofauti na mipangilio) utaratibu wa uchaguzi hauna umuhimu. Katika matukio madogo inawezekana kuhesabu idadi ya uchanganyiko.<br><br>Tiger Algebra inahesabu idadi ya uchanganyiko ikionyesha wewe suluhisho hatua kwa hatua. Kuanzisha, weka input yako kwa mojawapo wa fomu zifuatazo: <div style="white-space: normal;"> <p>Katika combinatorics, uchanganyiko ni uchaguzi wa vitu kutoka kwenye seti kubwa, ambapo utaratibu wa uchaguzi hauna umuhimu. Uchanganyiko hutumika mara nyingi kuhesabu idadi ya njia za kuchagua subset ya vitu kutoka seti kubwa.</p> <h2>Formula</h2> <p>Idadi ya uchanganyiko wa <math>k</math> vitu vilivyochaguliwa kutoka kwenye seti ya <math>n</math> vitu (inajulikana kama <math>C(n, k)</math> au <math>{n \choose k}</math>) ina hesabiwa kutumia formula ya uchanganyiko:</p> <div style="margin-left: 20px;"> <math>C(n, k) = \frac{n!}{k!(n - k)!}</math>, </div> <p>ambapo <math>n!</math> inawakilisha factorial ya <math>n</math>, imefafanuliwa kama bidhaa ya integers chanya zote chini au sawa na <math>n</math>.</p> <h2>Sifa za pekee</h2> <ul> <li>Idadi ya uchanganyiko ni integer ambayo sio hasi kila wakati.</li> <li>Uchanganyiko ni bila mpangilio, hii ina maanisha kuwa uchaguzi wa seti ile ile ya vitu kwa utaratibu tofauti hauzalishi uchanganyiko mpya.</li> <li>Idadi ya uchanganyiko mara nyingi hutumika katika hesabu za uwezekano na matatizo ya kuhesabu.</li> </ul> <h2>Example</h2> <p>Chukulia tume na seti ya herufi 5: A, B, C, D, na E. Tunataka kuchagua herufi 3 kutoka kwenye seti hii bila kujali utaratibu wa uchaguzi. Idadi ya uchanganyiko inayowezekana ni:</p> <div style="margin-left: 20px;"> <math>C(5, 3) = \frac{5!}{3!(5 - 3)!} = \frac{5 \times 4 \times 3}{3 \times 2 \times 1} = 10</math>. </div> <p>Kwa hivyo, kuna uchanganyiko 10 tofauti wa herufi 3 ambazo zinaweza kuchaguliwa kutoka kwenye seti {A, B, C, D, E}.</p> <p>Uchanganyiko ni msingi katika hisabati combinatorial na una matumizi mengi katika maeneo mbalimbali, ikiwa ni pamoja na takwimu, sayansi ya kompyuta, na utarakilishi.</p> </div>
DistanceBetweenTwoPointsAndTheirMidpoint	Distance between two points	Umbali kati ya nukta mbili
DistanceBetweenTwoPointsAndTheirMidpointContent	The distance formula, an application of the Pythagorean theorem, is a very helpful tool for finding the distance between two points. The Pythagorean theorem states the following: in a right triangle, the length of side <math>a</math> squared plus the length of side <math>b</math> squared is equal to the length of the hypotenuse (side <math>c</math>) squared.<br> <math>a^2+b^2=c^2</math><br> <img src="/images/distance-between-two-points-graph1.png" alt="Distance between two points graph"><br><br> The hypotenuse (<math>c</math>) is the longest side of a right triangle and is always opposite the right angle. The length of the hypotenuse also represents the distance between points <b>A</b> and <b>B</b>, which can each be represented by two coordinates: an <math>x</math> coordinate and a <math>y</math> coordinate.<br> Point <b>A</b> = <math>(x_1,y_1)</math><br> Point <b>B</b> = <math>(x_2,y_2)</math><br><br> To get the distance formula, we can rewrite the Pythagorean theorem as:<br> <math>d=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}</math><br> in which <math>d</math> represents the distance between points <b>A</b> and <b>B</b>, and the Xs and Ys represent the <math>x</math> and <math>y</math> coordinates of points <b>A</b> and <b>B</b>.<br><br> In order to find the distance between two points, enter their coordinates (for example (1,2) and (3,4)) and click the solve button.	Fomula ya umbali, ambayo ni utekelezaji wa theorem ya Pythagoras, ni zana muhimu sana kwa kupata umbali kati ya nukta mbili. Theorem ya Pythagoras inaeleza ifuatavyo: katika pembe tatu ya kulia, urefu wa upande <math>a</math> ukisquarishwa pamoja na urefu wa upande <math>b</math> ukisquarishwa unalingana na urefu wa hypotenuse (upande <math>c</math>) uliosquarishwa.<br> <math>a^2+b^2=c^2</math><br> <img src="/images/distance-between-two-points-graph1.png" alt="graph ya umbali kati ya nukta mbili"><br><br> Hypotenuse (<math>c</math>) ni upande mrefu zaidi wa pembe tatu ya kulia na daima ni kinyume na pembe ya kulia. Urefu wa hypotenuse pia unawakilisha umbali kati ya alama <b>A</b> na <b>B</b>, ambazo kila moja inaweza kuwakilishwa na viambishi vya <math>x</math> na <math>y</math>.<br> Nukta <b>A</b> = <math>(x_1,y_1)</math><br> Nukta <b>B</b> = <math>(x_2,y_2)</math><br><br> Ili kupata formula ya umbali, tunaweza kuandika upya theorem ya Pythagoras kama:<br> <math>d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</math><br> ambapo <math>d</math> inawakilisha umbali kati ya nukta <b>A</b> na <b>B</b>, na X na Y zinawakilisha viambishi <math>x</math> na <math>y</math> vya nukta <b>A</b> na <b>B</b>.<br><br> Ili kupata umbali kati ya nukta mbili, ingiza viambishi vyao (kwa mfano (1,2) na (3,4)) kisha bonyeza kitufe cha kutatua.
DividingExponents	Dividing exponents	Kugawanya nguvu za idadi
DividingExponentsContent	<div style="white-space: normal;"> <p>When dividing expressions with exponents, you can use the properties of exponents to simplify the expression. Here's how:</p> <h2>Step 1: Subtract Exponents</h2> <p>To divide two terms with the same base, subtract the exponent of the divisor from the exponent of the dividend. For example, <math>x^a ÷ x^b = x^(a - b)</math>.</p> <p>For instance, if you have <math>x^5 ÷ x^2</math>, you subtract the exponents: <math>x^(5 - 2) = x^3</math>.</p> <h2>Step 2: Simplify</h2> <p>If there are additional terms in the expression, simplify them as well. For example, if you have <math>(x^5 * y^3) ÷ (x^2 * y)</math>, divide the x's and y's separately: <math>x^(5 - 2) * y^(3 - 1) = x^3 * y^2</math>.</p> <p>Remember, when dividing exponents with the same base, subtract the exponents.</p> <p>This process is crucial in algebra and higher-level mathematics when dealing with expressions containing exponents.</p> </div>	<div style='white-space: normal;'> <p>Unapogawanya maneno yenye nguvu, unaweza kutumia sifa za nguvu kusawazisha kisawe. Hivi ndivyo unavyofanya:</p> <h2>Hatua ya 1: Toa Nguvu</h2> <p>Kugawanya maneno mawili na msingi sawa, toa nguvu ya divisor kutoka kwa nguvu ya dividend. Kwa mfano, <math>x^a ÷ x^b = x^(a - b)</math>.</p> <p>Kwa mfano, ikiwa una <math>x^5 ÷ x^2</math>, toa nguvu: <math>x^(5 - 2) = x^3</math>.</p> <h2>Hatua ya 2: Sawazisha</h2> <p>Ikiwa kuna maneno zaidi katika kisawe, sawazisha pia. Kwa mfano, ikiwa una <math>(x^5 * y^3) ÷ (x^2 * y)</math>, gawanya x na y tofauti: <math>x^(5 - 2) * y^(3 - 1) = x^3 * y^2</math>.</p> <p>Kumbuka, wakati unagawanya nguvu za idadi zenye msingi sawa, toa nguvu.</p> <p>Mchakato huu ni muhimu katika algebra na hesabu ya juu wakati unafanya kazi na maneno yenye nguvu.</p> </div>
DividingFractions	Dividing fractions	Kugawanya sehemu
DividingFractionsContent	<div style="white-space: normal;"> <p>Dividing fractions is a fundamental operation in mathematics, especially in arithmetic and algebra. When dividing fractions, we follow specific rules to simplify the process and obtain the result.</p> <h2>Basic Concepts</h2> <p>To divide fractions, it's important to understand the following concepts:</p> <ul> <li><strong>Fractions:</strong> Numbers that represent parts of a whole or ratios of two quantities.</li> <li><strong>Dividing fractions:</strong> The process of dividing one fraction by another to find the quotient.</li> <li><strong>Reciprocal:</strong> The reciprocal of a fraction <math>\frac{a}{b}</math> is <math>\frac{b}{a}</math>.</li> </ul> <h2>Division Techniques</h2> <p>There are two common techniques to divide fractions:</p> <ul> <li><strong>Using reciprocals:</strong> Multiply the first fraction by the reciprocal of the second fraction.</li> <li><strong>Converting to multiplication:</strong> Convert the division operation to multiplication by multiplying the first fraction by the reciprocal of the second fraction.</li> </ul> <h2>Examples</h2> <p>Let's consider a few examples to illustrate the division of fractions:</p> <h3>Example 1:</h3> <p>Divide <math>\frac{3}{4}</math> by <math>\frac{1}{2}</math></p> <p>Using the reciprocal technique, we multiply the first fraction by the reciprocal of the second fraction:</p> <p><math>\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{6}{4}</math></p> <p>Since <math>\frac{6}{4}</math> can be simplified to <math>\frac{3}{2}</math>, the result is <math>\frac{3}{2}</math>.</p> <h3>Example 2:</h3> <p>Divide <math>\frac{5}{8}</math> by <math>\frac{2}{3}</math></p> <p>We convert the division operation to multiplication by multiplying the first fraction by the reciprocal of the second fraction:</p> <p><math>\frac{5}{8} \div \frac{2}{3} = \frac{5}{8} \times \frac{3}{2}</math></p> <p>After simplification, we get the result as <math>\frac{15}{16}</math>.</p> <h2>Conclusion</h2> <p>Dividing fractions is a fundamental operation that plays a crucial role in various mathematical contexts. Mastering the techniques of fraction division enables efficient problem-solving and enhances mathematical understanding.</p> </div>	<div style="white-space: normal;"> <p>Kugawanya sehemu ni operesheni muhimu sana katika hisabati, hasa katika gezgezana na aljebra. Tunapogawanya sehemu, tunafuata sheria maalum kufanya mchakato kuwa rahisi na kupata matokeo.</p> <h2>Dhanna Muhimu</h2> <p>Kugawanya sehemu, ni muhimu kuelewa dhana zifuatazo:</p> <ul> <li><strong>Sehemu:</strong> Hesabu zinazowakilisha vipande vya jambo lote au uwiano wa kiasi viwili.</li> <li><strong>Kugawanya sehemu:</strong> Mchakato wa kugawanya sehemu moja na sehemu nyingine kupata majumuisho.</li> <li><strong>Mpangiliocha:</strong> Mpangiliocha wa sehemu <math>\frac{a}{b}</math> ni <math>\frac{b}{a}</math>.</li> </ul> <h2>Mbinu za Kugawanya</h2> <p>Zipo mbinu mbili za kawaida za kugawanya sehemu:</p> <ul> <li><strong>Kutumia mpangiliocha:</strong> Zidisha sehemu ya kwanza na mpangiliocha wa sehemu ya pili.</li> <li><strong>Kubadilisha kuwa kuzidisha:</strong> Badilisha operesheni ya kugawanya kuwa kuzidisha kwa kuzidisha sehemu ya kwanza na mpangiliocha wa sehemu ya pili.</li> </ul> <h2>Mifano</h2> <p>Acheni tuchukue mifano michache ili kuelezea ugawaji wa sehemu:</p> <h3>Mfano 1:</h3> <p>Gawa <math>\frac{3}{4}</math> kwa <math>\frac{1}{2}</math>.</p> <p>Kwa kutumia mbinu ya mpangiliocha, zidisha sehemu ya kwanza na mpangiliocha wa sehemu ya pili:</p> <p><math>\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{6}{4}</math></p> <p>Maana yake <math>\frac{6}{4}</math> inaweza kusimplishwa kuwa <math>\frac{3}{2}</math>, matokeo yake ni <math>\frac{3}{2}</math>.</p> <h3>Mfano 2:</h3> <p>Gawa <math>\frac{5}{8}</math> kwa <math>\frac{2}{3}</math>.</p> <p>Tunabadilisha operesheni ya kugawanya kuwa kuzidisha kwa kuzidisha sehemu ya kwanza na mpangiliocha wa sehemu ya pili:</p> <p><math>\frac{5}{8} \div \frac{2}{3} = \frac{5}{8} \times \frac{3}{2}</math></p> <p>Baada ya kusawazisha, tunapata matokeo kama <math>\frac{15}{16}</math>.</p> <h2>Hitimisho</h2> <p>Kugawanya sehemu ni operesheni muhimu inayotoa mchango muhimu katika muktadha tofauti wa hisabati. Kujifunza mbinu za ugawaji wa sehemu husababisha utatuzi wa matatizo kuwa rahisi na kukuza uelewa wa hisabati.</p> </div>
EquationswhichareReducibletoQuadratic	Equations which are reducible to quadratic	Mafungu ambayo yanaweza kupunguzwa kuwa ya pili
EquationswhichareReducibletoQuadraticContent	The Tiger Algebra solver shows you, step by step how to find which equations are reducable to which Quadratic equations. <div style="white-space: normal;"> <p>Some equations can be transformed or manipulated into a quadratic equation, which makes them easier to solve. Equations that can be reduced to quadratic form often involve variables raised to powers or contain multiple terms.</p> <h2>Types of Equations</h2> <p>Equations that are reducible to quadratic form include:</p> <ul> <li><strong>Equations involving radicals:</strong> Equations with square roots or other radicals can often be squared to eliminate the radical and form a quadratic equation.</li> <li><strong>Equations with rational expressions:</strong> Equations containing rational expressions can sometimes be transformed into quadratic equations by clearing denominators.</li> <li><strong>Equations with variable substitutions:</strong> Substituting a new variable can sometimes transform a given equation into a quadratic equation.</li> <li><strong>Equations involving trigonometric functions:</strong> Trigonometric identities or trigonometric substitutions can sometimes reduce trigonometric equations to quadratic form.</li> </ul> <h2>Example</h2> <p>Let's consider the equation <math>\frac{1}{x} - \frac{1}{y} = 3</math>. We can rewrite it as:</p> <div style="margin-left: 20px;"> <math>\frac{1}{x} - \frac{1}{y} = 3 \Rightarrow \frac{y - x}{xy} = 3 \Rightarrow y - x = 3xy</math>.<br> </div> <p>Now, let's make a substitution, <math>z = xy</math>. This transforms the equation into a quadratic equation:</p> <div style="margin-left: 20px;"> <math>y - x = 3xy \Rightarrow y - x = 3z \Rightarrow 3z - y + x = 0</math>.<br> </div> <p>This equation is now in quadratic form, and we can solve it using quadratic equation-solving techniques.</p> <p>Understanding how to manipulate equations into quadratic form can simplify the solving process and make it more manageable.</p> </div>	Mfadhili wa hesabu ya Tiger Algebra inakuonyesha, hatua kwa hatua jinsi ya kuona ni mafungu gani yanayoweza kupunguzwa kuwa yaongezeko ya mpangilio wa pili. <div style="white-space: normal;"> <p>Baadhi ya mafungu yanaweza kubadilishwa au kudhibitiwa kuwa yaongezeko ya mpangilio wa pili, jambo ambalo linawafanya kuwa rahisi kuyatatua. Mafungu ambayo yanaweza kupunguzwa kuwa yaongezeko ya mpangilio wa pili mara nyingi yatajumuisha viwango vinavyoongezwa kwa nguvu au yana mihimili mingi.</p> <h2>Aina za Mafungu</h2> <p>Mafungu yanayoweza kuelezwa na muundo wa pili awa ni:</p> <ul> <li><strong>Mafungu yanayohusisha mizizi:</strong> Mafungu yenye mizizi ya mraba au mizizi mingine mara nyingi yanaweza kuongezwa nguvu mbili ili kutoa mizizi na kuunda yaongezeko ya mpangilio wa pili.</li> <li><strong>Mafungu yenye misemo ya kimantiki:</strong> Mafungu yanayojumuisha misemo ya kimantiki mara nyingi yanaweza kupangwa kuwa yaongezeko ya mpangilio wa pili kwa kuiosha wabebaji.</li> <li><strong>Mafungu yenye mabadilishano ya viwango:</strong> Kuchukua viwango vipya mara nyingi huweza kubadilisha mafungu yaliyopo kuwa ya yaongezeko ya mpangilio wa pili.</li> <li><strong>Mafungu yanayojumuisha utendakazi wa trigonometri:</strong> Utambulisho wa trigonometri au mabadilishano ya trigonometri mara nyingi yanaweza kupunguza mafungu ya trigonometri kuwa yaongezeko ya mpangilio wa pili.</li> </ul> <h2>Mfano</h2> <p>Tuchunguze mafungu <math>\frac{1}{x} - \frac{1}{y} = 3</math>. Tunaweza kuandika tena ni:</p> <div style="margin-left: 20px;"> <math>\frac{1}{x} - \frac{1}{y} = 3 \Rightarrow \frac{y - x}{xy} = 3 \Rightarrow y - x = 3xy</math>.<br> </div> <p>Sasa, tufanye mabadilishano, <math>z = xy</math>. Hii inabadilisha mafungu kuwa ya yaongezeko ya mpangilio wa pili:</p> <div style="margin-left: 20px;"> <math>y - x = 3xy \Rightarrow y - x = 3z \Rightarrow 3z - y + x = 0</math>.<br> </div> <p>mafungu haya sasa yameandikwa na muundo wa pili, na tunaweza kuyatatua kwa kutumia utendakazi wa kutatua ya yaongezeko ya mpangilio wa pili.</p> <p>Kuelewa jinsi ya kudhibiti mafungu kuwa muundo wa pili kunaweza kupelekea mchakato wa kutatua kuwa rahisi na wa kufaa.</p> </div>
FactoringBinomialsAsSumOrDifferenceOfCubes	Factoring binomials as sum or difference of cubes	Kufafanua binomials kama jumla au tofauti ya vikuba
FactoringBinomialsAsSumOrDifferenceOfCubesContent	Factoring binomials as the sum or the difference of cubes <br><br> To find the sum or the difference of cubes, you have to apply one of two factoring formulas. They are almost the same, with one small difference: the placement of the minus sign. <br><br> Formula for the sum of cubes: <br> <math>a^3 + b^3 = (a + b)(a^2 – ab + b^2)</math> <br><br> Formula for the difference of cubes: <br> <math>a^3 – b^3 = (a – b)(a^2 + ab + b^2)</math> <br><br> In the formula for factoring the sum of cubes, the minus sign is found in the quadratic factor: <math>a^2 – ab + b^2</math>. In the one used to factor the difference of cubes, the minus sign is located in the linear factor: <math>a – b</math>. <br><br> Be sure to apply the appropriate factoring formula! <br><br> Enter the binomial into the calculator and Tiger will show you, step by step, how to solve it.	Kufafanua binomials kama jumla au tofauti ya vikuba <br><br> Ili kupata jumla au tofauti ya vikuba, unahitaji kutumia moja ya fomula mbili za kufafanua. Zote ni karibu sawa, na tofauti ndogo: nafasi ya ishara ya kinyume. <br><br> Fomula ya jumla ya vikuba: <br> <math>a^3 + b^3 = (a + b)(a^2 – ab + b^2)</math> <br><br> Fomula kwa tofauti ya vikuba: <br> <math>a^3 – b^3 = (a – b)(a^2 + ab + b^2)</math> <br><br> Katika fomula ya kufafanua jumla ya vikuba, ishara ya kinyume inapatikana katika factor ya quadratic: <math>a^2 – ab + b^2</math>. Katika hiyo ambayo hutumiwa kufafanua tofauti ya vikuba, ishara ya kinyume iko katika factor ya linear: <math>a – b</math>. <br><br> Hakikisha unatumia fomula inayofaa ya kufafanua! <br><br> Andika binomial katika kalkulata na Tiger itakuonyesha, hatua kwa hatua, jinsi ya kuipatia suluhisho.
FactoringBinomialsasDifferenceofSquares	Factoring binomials as difference of squares	Kufafanua binomials kama tofauti ya mraba
FactoringBinomialsasDifferenceofSquaresContent	A binomial is factorable only if it is one of three things a Difference of Squares, a Difference of Cubes, or a Sum of Cubes. A binomial is a Difference of Squares if both terms are perfect squares. Recall we may have to factor out a common factor first. <br><br> If we determine that a binomial is a difference of squares, we factor it into two binomials. The first being the square root of the first term minus the square root of the second term. The second being the square root of the first term plus the square root of the second term. <div style="white-space: normal;"> <p>A binomial is an algebraic expression consisting of two terms. The difference of squares is a special case of factoring where a binomial can be factored into the product of two binomials.</p> <p>The formula for factoring a binomial as the difference of squares is: <math>a^2 - b^2 = (a + b)(a - b)</math>.</p> <h2>Step-by-Step Process</h2> <p>To factor a binomial as the difference of squares, follow these steps:</p> <ol> <li>Identify the square of each term in the binomial.</li> <li>Write the binomial as the difference of squares using the formula above.</li> <li>Factor the expression if possible.</li> </ol> <h2>Example</h2> <p>Let's factor the binomial <math>x^2 - 9</math> as the difference of squares:</p> <p>Step 1: Identify the squares - <math>x^2</math> and <math>9</math> are both perfect squares.</p> <p>Step 2: Write the difference of squares - <math>x^2 - 9 = (x + 3)(x - 3)</math>.</p> <p>Step 3: The expression is now factored as the difference of squares.</p> <p>Factoring binomials as the difference of squares is a fundamental technique in algebra and is often used in various mathematical problems.</p> </div>	Binomial inaweza kufafanuliwa tu ikiwa ni mojawapo ya vitu vitatu Tofauti ya Mraba, Tofauti ya Vikuba, au Jumla ya Vikuba. Binomial ni Tofauti ya Mraba ikiwa vigezo vyote ni mraba kamili. Kumbuka kwamba tunaweza haja ya kufafanua factor ya kawaida kwanza. <br><br> Ikiwa tunaamua kwamba binomial ni tofauti ya mraba, tunaifafanua kwenye binomial mbili. Ya kwanza ikiwa ni mzizi wa kipekee wa kigezo cha kwanza minus mzizi wa kipekee wa kigezo cha pili. Ya pili ikiwa ni mzizi wa kipekee wa kigezo cha kwanza plus mzizi wa kipekee wa kigezo cha pili. <div style="white-space: normal;"> <p>Binomial ni usemi wa algebra unaokuwa na vigezo viwili. Tofauti ya mraba ni kesi maalum ya kufafanua ambapo binomial inaweza kufafanuliwa kwenye bidhaa ya binomials mbili.</p> <p>Fomula ya kufafanua binomial kama tofauti ya mraba ni: <math>a^2 - b^2 = (a + b)(a - b)</math>.</p> <h2>Mchakato wa Hatua kwa Hatua</h2> <p>Kufafanua binomial kama tofauti ya mraba, fuata hatua hizi:</p> <ol> <li>Tambua mraba wa kila kigezo cha binomial.</li> <li>Andika binomial kama tofauti ya mraba kwa kutumia fomula hapo juu.</li> <li>Fafanua usemi ikiwa inawezekana.</li> </ol> <h2>Mfano</h2> <p>Acha tufafanue binomial <math>x^2 - 9</math> kama tofauti ya mraba:</p> <p>Hatua 1: Tambua mraba - <math>x^2</math> na <math>9</math> ni mraba kamili.</p> <p>Hatua 2: Andika tofauti ya mraba - <math>x^2 - 9 = (x + 3)(x - 3)</math>.</p> <p>Hatua 3: Sasa usemi umeafanuliwa kama tofauti ya mraba.</p> <p>Kufafanua binomials kama tofauti ya mraba ni tekniki muhimu katika algebra na mara nyingi hutumika katika matatizo ya hesabu mbalimbali.</p> </div>

Key	English	Swahili
ChemistryFindingAnAtomicWeightStep2TextUnit2	{Step2ChemicalFormula} molecules consist of: <br> {Step2AllAtomsList:{Element} {Symbol}={Mass} u <br> \|}	Molekuli za {Step2ChemicalFormula} zinaundwa na: <br> {Step2AllAtomsList:{Element} {Symbol}={Mass} u <br> \|}
ChemistryFindingAnAtomicWeightStep2Title1	Identify the atomic mass of each element	Tambua uzito wa atomiki wa kila kipengele
ChemistryFindingAnAtomicWeightStep3TextUnit1	{Step3AllAtomsList:{ElementWithFrequency} → {FrequencyTimesMass} u <br>\|}	{Step3AllAtomsList:{ElementWithFrequency} → {FrequencyTimesMass} u <br>\|}
ChemistryFindingAnAtomicWeightStep3Title1	Calculate the total atomic mass of each element in a molecule of {Step3ChemicalFormula}	Hesabu jumla ya uzito wa atomiki wa kila kipengele kwenye molekuli ya {Step3ChemicalFormula}
ChemistryFindingAnAtomicWeightStep4TextUnit1	{Step4ResultExpended}	{Step4ResultExpended}
ChemistryFindingAnAtomicWeightStep4TextUnit2	The average molecular mass of {Step4ChemicalFormula} is {Step4Result} u.	Masi ya wastani ya molekuli ya {Step4ChemicalFormula} ni {Step4Result} u.
ChemistryFindingAnAtomicWeightStep4Title1	Calculate the mass of the molecule {Step4ChemicalFormula}	Hesabu uzito wa molekuli {Step4ChemicalFormula}
ChemistryFindingAnAtomicWeightStep5Title1	Graph of molecular composition by atoms	Chati ya muundo wa molekuli kwa atomi
ChemistryFindingAnAtomicWeightStep6Title1	Graph of molecular composition by mass	Chati ya muundo wa molekuli kwa masi
ChemistryNumberOfAtoms	# of atoms	# ya atomi
ChemistrySymbol	Symbol	Ishara
Circle	Circles	Mzunguko
CircleArea	Area	Eneo
CircleCenter	Center	Kitovu
CircleCircumference	Circumference (or perimeter)	Mzunguko (au perimeter)
CircleContent	In geometry, a circle is a shape made up of all the points on a plane at a fixed distance around a given point (the center). The equation for a circle is <math>(x-h)^2+(y-k)^2=r^2</math>, in which <math>h</math> and <math>k</math> represent the circle's center and <math>r</math> represents the circle's radius, the distance from the circle's center to any point on its perimeter. A circle with a center at <math>(4,5)</math> and a radius of <math>10</math>, for example, could be expressed as <math>(x-4)^2+(y-5)^2=100</math> <br> <img src="/images/circleImg1.png" alt="circle graph"> <br> <img src="/images/circleImg2.png" alt="circle related terms diameter, radius, chord, secant, tangent"> <br> <b>Relevant terms:</b><br> <ul> <li><b>Center</b>: The point around which a circle is formed. All points on a circle's perimeter are the same distance from the circle's center.</li><br> <li><b>Circumference</b>: The distance around a circle.</li><br> <li><b>Radius</b>: A line segment that lies between the center of a circle and any point on the circle's perimeter</li>.<br> <li><b>Diameter</b>: A line segment that lies between two points on a circle's perimeter and runs through the circle's center. It is equal to two times the circle's radius.</li><br> <li><b>Chord</b>: A line segment that lies between two points on a circle's perimeter and does not run through the circle's center.</li><br> <li><b>Secant</b>: A line that intersects two points on a circle's perimeter.</li><br> <li><b>Tangent</b>: A line that intersects one point on a circle's perimeter.</li> </ul>	Katika jiometri, mduara ni muundo uliotengeneza na idadi zote kwenye nafasi kwa umbali wa kudumu karibu na nafasi fulani (kitovu). Equation ya mduara ni <math>(x-h)^2+(y-k)^2=r^2</math>, ambako <math>h</math> na <math>k</math> zinawakilisha kitovu chake na <math>r</math> inawakilisha radiusi ya mduara, umbali kutoka kitovu cha mduara kwa nafasi yoyote kwenye mipaka yake. Mduara na kitovu kwenye <math>(4,5)</math> na radiusi ya <math>10</math>, kwa mfano, inaweza kuelezea kama <math>(x-4)^2+(y-5)^2=100</math><br> <img src="/images/circleImg1.png" alt="mchoro wa mduara"> <br> <img src="/images/circleImg2.png" alt="maneno yanayohusiana na mduara diameteri, radiusi, chord, secant, tangent"> <br> <b>Maneno yanayohusiana:</b><br> <ul> <li><b>Kitovu</b>: Nafasi ambayo mduara unajenga. Idadi zote kwenye mipaka ya mduara ziko umbali sawa kutoka kitovu cha mduara.</li><br> <li><b>Mzunguko</b>: Umbali kuzunguka mduara.</li><br> <li><b>Radiusi</b>: Sehemu ya mstari inayolala kati ya kitovu cha mduara na nafasi yoyote kwenye mipaka yake.</li><br> <li><b>Diameteri</b>: Sehemu ya mstari inayolala kati ya nafasi mbili kwenye mipaka yake na inapitia kitovu chake. Ni sawa na ukubwa wa maredio mara mbili.</li><br> <li><b>Chord</b>: Sehemu ya mstari inayolala kati ya nafasi mbili kwenye mipaka ya mduara na haipiti katikati ya mduara.</li><br> <li><b>Secant</b>: Line inayokatisha nafasi mbili kwenye mipaka ya mduara.</li><br> <li><b>Tangent</b>: Mstari unaokatisha nafasi moja kwenye mipaka ya mduara.</li> </ul>
CircleDiameter	Diameter	Kipenyo
CircleFormattingGuide	<h3>Description</h3><p>In geometry, a circle is a shape made up of all the points on a plane at a fixed distance around a given point (the center). The equation for a circle is (x-h)2+(y-k)2=r2, in which h and k represent the circle's center and r represents the circle's radius, the distance from the circle's center to any point on its perimeter.</p> <h3>Format</h3><p>Try inputting the coordinates of a circle's center followed by its radius or diameter in the format "center (a, b) radius (c)" in which in which (a,b). You can also input the equation of a circle written in standard form.</p> <h3>Examples</h3> <ul> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/center(3,4)%20radius(3)/">center(3,4) radius(3)</a></li> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/center(3,4)%20diameter(6)/">center(3,4) diameter(6)</a></li> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/radius(3)%20center(3,4)/">radius(3) center(3,4)</a></li> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/diameter(6)%20center(3,4)/">diameter(6) center(3,4)</a></li> <li><a href="/en/solution/properties-of-circle/(x-3)%5E2+(y-4)%5E2=4/">(x-3)^2+(y-4)^2=4</a></li> </ul>	<h3>Maelezo</h3><p>Katika jiometri, duara ni umbo ambalo linajumuisha pointi zote kwenye eneo fulani lenye umbali uliothabiti kutokana na pointi moja (kitovu). Mfumo wa hesabu wa duara ni (x-h)2+(y-k)2=r2, ambapo h na k zinawakilisha kitovu cha duara na r inawakilisha radius ya duara, ambayo ni umbali kutoka kitovu cha duara hadi pointi yoyote kwenye mzunguko wake.</p> <h3>Muundo</h3><p>Jaribu kuingiza kipeo cha duara kifuatiwe na radius au kipenyo kwenye muundo wa "kitovu (a, b) radius (c)" ambapo (a,b). Pia unaweza kuingiza mfumo wa hesabu wa duara ulioandikwa kwenye muundo wa kawaida.</p> <h3>Mifano</h3> <ul> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/kitovu(3,4)%20radius(3)/">kitovu(3,4) radius(3)</a></li> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/kitovu(3,4)%20kipenyo(6)/">kitovu(3,4) kipenyo(6)</a></li> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/radius(3)%20kitovu(3,4)/">radius(3) kitovu(3,4)</a></li> <li><a href="/en/solution/properties-of-circles-from-center-point-and-radius-diameter/kipenyo(6)%20kitovu(3,4)/">kipenyo(6) kitovu(3,4)</a></li> <li><a href="/en/solution/properties-of-circle/(x-3)%5E2+(y-4)%5E2=4/">(x-3)^2+(y-4)^2=4</a></li> </ul>
CircleFromCenterPointAndRadius	Properties of circles from center point and radius/diameter	Sifa za duara kutoka kwa kituo na radius/diameter
CircleFromCenterPointAndRadiusContent	In geometry, a circle is a shape made up of all the points on a plane at a fixed distance around a given point (the center). The equation for a circle is <math>(x-h)^2+(y-k)^2=r^2</math>, in which <math>h</math> and <math>k</math> represent the circle's center and <math>r</math> represents the circle's radius, the distance from the circle's center to any point on its perimeter. A circle with a center at <math>(4,5)</math> and a radius of <math>10</math>, for example, could be expressed as <math>(x-4)^2+(y-5)^2=100</math> <br> <img src="/images/CircleFromCenterPointAndRadius_graph1.png" alt="circle graph"> <br> <img src=/images/solver/LinesInACircle.png alt=equation of a circle and it's lines> <br> <b>Relevant terms:</b><br> <ul> <li><b>Center</b>: The point around which a circle is formed. All points on a circle's perimeter are the same distance from the circle's center.</li><br> <li><b>Circumference</b>: The distance around a circle.</li><br> <li><b>Radius</b>: A line segment that lies between the center of a circle and any point on the circle's perimeter</li>.<br> <li><b>Diameter</b>: A line segment that lies between two points on a circle's perimeter and runs through the circle's center. It is equal to two times the circle's radius.</li><br> <li><b>Chord</b>: A line segment that lies between two points on a circle's perimeter and does not run through the circle's center.</li><br> <li><b>Secant</b>: A line that intersects two points on a circle's perimeter.</li><br> <li><b>Tangent</b>: A line that intersects one point on a circle's perimeter.</li> </ul>	Katika jiometri, duara ni umbo lililotengenezwa na vipengee vyote kwenye ndege kwa umbali uliowekwa karibu na kiwango fulani (kitovu). Equation ya duara ni <math>(x-h)^2+(y-k)^2=r^2</math>, ambayo <math>h</math> na <math>k</math> inawakilisha kitovu cha duara na <math>r</math> inawakilisha radisi ya duara, umbali kutoka kitovu cha duara kwenda kipengele chochote kwenye mzunguko wake. Duara lenye kitovu kwenye <math>(4,5)</math> na radisi ya <math>10</math>, kwa mfano, linaweza kuelezewa kama <math>(x-4)^2+(y-5)^2=100</math> <br> <img src="/images/CircleFromCenterPointAndRadius_graph1.png" alt="graph ya duara"> <br> <img src=/images/solver/LinesInACircle.png alt=equation ya duara na mistari yake> <br> <b>Istilahi zinazohusiana:</b><br> <ul> <li><b>Kitovu</b>: Kipengee ambacho duara kilisukumwa. Vipengee vyote kwenye mzunguko wa duara vinatoka kwa umbali sawa kutoka kitovu cha duara.</li><br> <li><b>Mzunguko</b>: Umbali karibu na duara.</li><br> <li><b>Radisi</b>: Segment ya line ambayo inakaa kati ya kitovu cha duara na kipengee chochote kwenye mzunguko wa duara</li>.<br> <li><b>Kipenyo</b>: Segment ya line ambayo inakaa kati ya vipengee viwili kwenye mzunguko wa duara na inapita kupitia kitovu cha duara. Inasawa na mara mbili ya radisi ya duara.</li><br> <li><b>Chord</b>: Segment ya line ambayo inakaa kati ya vipengee viwili kwenye mzunguko wa duara na haipiti kupitia kitovu cha duara.</li><br> <li><b>Secant</b>: Line ambayo inakata vipengee viwili kwenye mzunguko wa duara.</li><br> <li><b>Tangent</b>: Line ambayo inakata kipengee kimoja kwenye mzunguko wa duara.</li> </ul>
CircleFromCenterPointAndRadiusFinalResultTypeArea	Area	Eneo
CircleFromCenterPointAndRadiusFinalResultTypeCircumference	Circumference	Mzunguko
CircleFromCenterPointAndRadiusFinalResultTypeDiameter	Diameter	Kipenyo
CircleFromCenterPointAndRadiusFinalResultTypeGeneralEquation	Expanded form equation	Equation iliyopanuliwa
CircleFromCenterPointAndRadiusFinalResultTypeRadius	Radius	Radisi
CircleFromCenterPointAndRadiusFinalResultTypeStandardEquation	Standard form equation	Equation ya kawaida
CircleFromCenterPointAndRadius.plugin	properties of circles	sifa za duara
CircleFromCenterPointAndRadiusStep1TextUnit1	A circle's radius (<math>r</math>) is half the length of its diameter (<math>d</math>). To find the radius, plug <math>d</math> into the formula:	Radius ya duara (<math>r</math>) ni nusu ya urefu wa kipenyo chake (<math>d</math>). Kupata radius, weka <math>d</math> kwenye formula:
CircleFromCenterPointAndRadiusStep1TextUnit2	<span class="formula-look f-left"><math>r=d/2</math></span><br> <math>d={Step1d}</math><br> <math>r={Step1d}/2</math><br> <math>r={Step1r}</math>	<span class="formula-look f-left"><math>r=d/2</math></span><br> <math>d={Step1d}</math><br> <math>r={Step1d}/2</math><br> <math>r={Step1r}</math>
CircleFromCenterPointAndRadiusStep1Title1	Find the radius	Pata radius

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core/TigerAlgebra.Localization/Resources/TigerSharedResource.sw.resx, string 442