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Key	English	Spanish
Approximation	Approximation	Aproximación
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.	La aproximación en matemáticas se define como una cantidad parecida en valor, pero no igual a la cantidad deseada. Hay dos símbolos matemáticos que denotan una igualdad aproximada: un signo igual ondulado (≈) y un signo igual punteado (≒ o ≓). <br><br> La aproximación se usa a menudo cuando una fórmula precisa o un número entero no se conocen o son difíciles de determinar, o también con los números irracionales como π. La aproximación puede convertir un cálculo complejo, por ejemplo, que contenga decimales, en uno que utilice números enteros, facilitando así su resolución. Esto se consigue redondeando los valores decimales antes de realizar las demás operaciones.
ArithmeticSequence	Arithmetic sequences	Progresión aritmética
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.	Una secuencia, sucesión o progresión aritmética es un conjunto de números en los que la diferencia entre los términos consecutivos (términos que van uno tras otro) es constante. Por ejemplo, todos los términos consecutivos en la secuencia aritmética:<br><b> 1, 4, 7, 10, 13, 16, 19, . . .</b><br> tienen una diferencia común de 3.<br><br>Aunque es posible utilizar otras variables, por lo general, usamos las siguientes para representar los términos de una secuencia aritmética:<br> <b>a1</b> representa el primer término de la secuencia;<br> <b>an</b> representa el término enésimo (el término que estamos intentando averiguar);<br> <b>d</b> representa la distancia común entre los términos consecutivos;<br> <b>n</b> representa el número de términos en la secuencia.<br><br>Tiger identifica las secuencias aritméticas y muestra sus términos, la suma de sus términos y su formas explícita y recursiva.
BasicComplexOperations	Basic complex operations	Operaciones básicas con números complejos
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.	Un número complejo es aquel que puede expresarse en la forma <math>a + bi</math>, donde a y b son números reales e i es la unidad imaginaria, que satisface la ecuación <math>x^2=−1</math>, es decir, <math>i^2=−1</math>. En esta expresión, a es la parte real y b es la parte imaginaria del número complejo. <br><br> La calculadora de álgebra Tiger realiza las operaciones básicas de números complejos/imaginarios y muestra la solución paso a paso.
CancelingOut	Canceling out	Reducir fracciones
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.	Para simplificar fracciones, a veces se puede dividir la parte de arriba (NUMERADOR) y la parte de abajo (DENOMINADOR) por el mismo número. Esta simplificación se llama reducir una fracción. Los problemas más usuales requieren que escribas una fracción en sus términos más simples. Para conseguirlo, debes reducir progresivamente la fracción hasta que ya no pueda reducirse más. <br><br> Tiger Algebra puede ayudarte a comprobar tus respuestas cuando reduzcas una fracción.
CircleTopic	Circle from equation	Circunferencia a partir de ecuación
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>	En geometría, una circunferencia es una forma constituida por todos los puntos de un plano que se encuentran a una distancia fija de un punto definido (el centro). La ecuación para una circunferencia es <mah>((x-h)^2)+((y-k)^2)=r^2</math>, donde (h,k) representa el centro de la circunferencia y r el radio de la circunferencia, es decir, la distancia desde el centro de la circunferencia a cualquier punto de su perímetro. Por ejemplo, una circunferencia con su centro en (4,5) y un radio de 10, podría expresarse como </mah>((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> Términos importantes: <ul> <li>Centro: el punto alrededor del cual se forma una circunferencia. Todos los puntos del perímetro de una circunferencia están a la misma distancia del centro de la misma.</li> <li>Perímetro: la distancia alrededor de una circunferencia.</li> <li>Radio: segmento de recta entre el centro de una circunferencia y cualquiera de los puntos de su perímetro.</li> <li>Diámetro: segmento de recta entre dos puntos del perímetro de una circunferencia pasando por el centro de la circunferencia. Equivale a dos veces el radio.</li> <li>Cuerda: segmento de recta entre dos puntos del perímetro de una circunferencia sin pasar por el centro de esta.</li> <li>Secante: recta que corta dos puntos del perímetro de una circunferencia.</li> <li>Tangente: recta que corta un punto del perímetro de una circunferencia.</li> </ul>
Combination	Combination	Combinaciones
CombinationTopic	Combination	Combinaciones
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>	En matemáticas, una combinación es una forma de seleccionar elementos de un conjunto en la que (a diferencia de las permutaciones) el orden de selección no importa. En problemas pequeños es posible calcular el número de combinaciones. <br><br> Tiger Algebra calcula el número de combinaciones y te muestra la solución paso a paso. Para activarlo, introduce tus datos de partida en una de las fórmulas siguientes:
DistanceBetweenTwoPointsAndTheirMidpoint	Distance between two points	Distancia entre dos puntos y su punto equidistante
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.	La fórmula de la distancia, una aplicación del teorema de Pitágoras, es una herramienta muy útil para averiguar la distancia entre dos puntos. El teorema de Pitágoras dice lo siguiente: en un triángulo rectángulo, la longitud del lado <math>a</math> al cuadrado más la longitud del lado <math>b</math> al cuadrado es igual a la longitud de la hipotenusa (lado <math>c</math>) al cuadrado.<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> La hipotenusa (<math>c</math>) es el lado más largo de un triángulo rectángulo y siempre está en el lado opuesto al ángulo recto. La longitud de la hipotenusa también representa la distancia entre los puntos <b>A</b> y <b>B</b>, los cuales se pueden representar cada uno con dos coordenadas: una coordenada <math>x</math> y una coordenada <math>y</math>.<br> Punto <b>A</b> = <math>(x_1,y_1)</math><br> Punto <b>B</b> = <math>(x_2,y_2)</math><br><br> Para obtener la fórmula de la distancia, podemos reescribir el teorema de Pitágoras como:<br> <math>d=sqrt((x_2−x_1)^2+(y_2−y_1)^2)</math><br> en la cual <math>d</math> representa la distancia entre los puntos <b>A</b> y <b>B</b>, y Xs e Ys representan las coordenadas <math>x</math> e <math>y</math> de los puntos <b>A</b> y <b>B</b> (sqrt significa raíz cuadrada).<br><br> Para poder calcular la distancia entre dos puntos, introduce sus coordenadas (por ejemplo (1,2) y (3,4)) y haz clic en el botón Resolver.
DividingExponents	Dividing exponents	Dividir exponentes
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>	Tiger Algebra te muestra paso a paso cómo dividir exponentes. Por ejemplo:
DividingFractions	Dividing fractions	Dividir fracciones
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>	La calculadora de álgebra Tiger divide las fracciones y te muestra la solución paso a paso, p. ej.:
EquationswhichareReducibletoQuadratic	Equations which are reducible to quadratic	Ecuaciones que pueden reducirse a cuadráticas
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>	Tiger Algebra te muestra paso a paso cómo averiguar qué ecuaciones pueden reducirse a ecuaciones de segundo grado y, en ese caso, a cuáles. Simplemente, introduce tu ecuación como en los ejemplos siguientes:
FactoringBinomialsAsSumOrDifferenceOfCubes	Factoring binomials as sum or difference of cubes	Factorizar binomios como suma o diferencia de cubos
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.	Factorizar binomios como la suma o la diferencia de cubos <br><br> Para averiguar la suma o la diferencia de cubos, tienes que aplicar una de las dos fórmulas de factorización. Son casi iguales salvo una pequeña diferencia: la colocación del signo menos. <br><br> Fórmula para la suma de cubos: <br> <math>a^3 + b^3 = (a + b)(a^2 – ab + b^2)</math> <br><br> Fórmula para la resta de cubos: <br> <math>a^3 – b^3 = (a – b)(a^2 + ab + b^2)</math> <br><br> En la fórmula para factorizar la suma de cubos, el signo menos se encuentra en el factor cuadrático: <math>a^2 – ab + b^2</math>. En cambio, en la que se usa para factorizar la diferencia de cubos, el signo menos está en el factor lineal: <math>a – b</math>. <br><br> ¡Asegúrate de aplicar la fórmula de factorización adecuada! <br><br> Introduce el binomio en la calculadora y Tiger te mostrará cómo resolverlo paso a paso.
FactoringBinomialsasDifferenceofSquares	Factoring binomials as difference of squares	Factorizar binomios como diferencia de cuadrados
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>	Un binomio solo se puede factorizar si es una de estas tres cosas: una diferencia de cuadrados, una diferencia de cubos o una suma de cubos. Un binomio es una diferencia de cuadrados si ambos términos son cuadrados perfectos. Ten en cuenta que puede que antes tengamos que descomponerlo sacando un factor común. <br><br> Si determinamos que un binomio es una diferencia de cuadrados, lo factorizamos en dos binomios. El primero es la raíz cuadrada del primer término menos la raíz cuadrada del segundo término. El segundo binomio es la raíz cuadrada del primer término más la raíz cuadrada del segundo término.
FactoringMultiVariablePolynomials	Factoring multivariable polynomials	Factorizar polinomios de varias variables
FactoringMultiVariablePolynomialsContent	<div style="white-space: normal;"> <p>Factoring multivariable polynomials involves finding the common factors among the terms of the polynomial and expressing it as the product of simpler polynomials.</p> <h2>Step-by-Step Process</h2> <p>To factor a multivariable polynomial, follow these steps:</p> <ol> <li>Identify common factors among the terms of the polynomial.</li> <li>Factor out the greatest common factor (GCF) from each term.</li> <li>Express the polynomial as the product of the GCF and the remaining factors.</li> <li>If possible, further factor the remaining polynomial using other factoring techniques such as grouping, difference of squares, or trinomial factoring.</li> </ol> <h2>Example</h2> <p>Let's factor the multivariable polynomial <math>2x^2y + 4xy^2</math>:</p> <p>Step 1: Identify common factors - The common factor among the terms is <math>2xy</math>.</p> <p>Step 2: Factor out the GCF - <math>2xy( x + 2y )</math>.</p> <p>Step 3: The polynomial is now factored as <math>2xy( x + 2y )</math>.</p> <p>Factoring multivariable polynomials is essential in algebra and other areas of mathematics for simplifying expressions and solving equations.</p> </div>	La calculadora de álgebra Tiger te enseña paso a paso cómo factorizar los polinomios de varias variables.
Factorial	Factorials	Factoriales
FactorialTopic	Factorials	Factoriales
FactorialContent	The factorial symbol ! is placed after a positive integer to indicate that it should be multiplied by every whole number less than it, through <math>1</math>. For example:<br> <math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math><br><br> The factorial operation, denoted as n!, states that:<br> <math>n! = n(n–1)(n–2)(n–3) · · · 3 · 2 · 1</math><br><br> The factorial operation is often used to mathematically express combinations, different ways that selected elements can be arranged when the order does not matter, and permutations, different ways that selected elements can be arranged when the order does matter.<br><br> The factorial of <math>0</math> is <math>1</math>.<br> <math>0! = 1</math>	El símbolo de factorial “!” detrás de un número entero positivo indica que debe multiplicarse por cada entero positivo menor que él hasta el <math>1</math>. Por ejemplo:<br> <math>6! = 6\cdot5\cdot4\cdot3\cdot2\cdot1</math><br><br> La operación factorial, que se representa como n!, establece que:<br> <math>n! = n(n–1)(n–2)(n–3) · · · 3 · 2 · 1</math><br><br> La operación factorial se usa a menudo para expresar matemáticamente combinaciones, es decir, las diferentes posibilidades de combinar elementos seleccionados cuando el orden no importa, y permutaciones, que son las diferentes posibilidades de combinar elementos seleccionados cuando el orden sí que importa.<br><br> El factorial de <math>0</math> es <math>1</math>.<br> <math>0! = 1</math>
FactoringPolynomialsHavingMoreThan3Terms	Factoring polynomials with four or more terms	Factorizar polinomios con cuatro o más términos

Key	English	Spanish
DistanceLine	d	d
DistanceNTX1	{Step1Point1X1Equals}	{Step1Point1X1Equals}
DistanceNTX2	{Step1Point1X2Equals}	{Step1Point1X2Equals}
DistanceNTY1	{Step1Point1Y1Equals}	{Step1Point1Y1Equals}
DistanceNTY2	{Step1Point1Y2Equals}	{Step1Point1Y2Equals}
DistanceP1	P1	P1
DistanceP2	P2	P2
distanceResultType	The distance between the 2 points is:	La distancia entre los 2 puntos es:
DistanceSolution	And so, the solution is {Step1Solution}	Y así, la solución es {Step1Solution}
DistanceSolveRadical	Solve the problem under the radical.	Resolver el problema bajo el radical.
DistanceSolveSQRT	Find the square root under the radical.	Averiguar la raíz cuadrada bajo el radical.
DistanceStep1TextUnit1	The coordinates of Point 1 are:<br> <math>X_1= </math>{Step1Point1X}<br> and <math>Y_1= </math>{Step1Point1Y}<br>	Las coordenadas del punto 1 son:<br> <math>X_1= </math>{Step1Point1X}<br> y <math>Y_1= </math>{Step1Point1Y}<br>
DistanceStep1TextUnit2	The coordinates of Point 2 are:<br> <math>X_2= </math>{Step1Point2X}<br> and <math>Y_2= </math>{Step1Point2Y}<br>	Las coordenadas del punto 2 son:<br> <math>X_2= </math>{Step1Point2X}<br> y <math>Y_2= </math>{Step1Point2Y}<br>
DistanceStep1TextUnit3	<b>d</b> is the distance between the two points.	<b>d</b> es la distancia entre los dos puntos.
DistanceStep1Title	Plot the two points and calculate the distance using the formula	Trazar los dos puntos y calcular la distancia utilizando la fórmula
DividingExponents	Dividing exponents	Dividir exponentes
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>	Tiger Algebra te muestra paso a paso cómo dividir exponentes. Por ejemplo:
DividingFractions	Dividing fractions	Dividir fracciones
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>	La calculadora de álgebra Tiger divide las fracciones y te muestra la solución paso a paso, p. ej.:
Domain	Domain	Dominio
DomainRangeAndRelationFromOrderedPairsTopicContent	In order to find the domain and range of ordered pairs and determine whether they are functions, we need to first understand a few key concepts.<br><br> <b>Ordered pair</b><br> An ordered pair is represented by two numbers that are written in a specific order, usually in parentheses, with a comma between them. For example: <math>(4, -5)</math> is an ordered pair. On an xy-plane—also called a Cartesian plane—an ordered pair represents the location of a point, where the first number in parentheses is the x-coordinate of the point and the second number is the y-coordinate of the point: <math>(x,y)</math>. In this case: <math>x=4, y=-5</math><br> To map these coordinates on a plane, we move outwards from the plane's origin <math>(0,0)</math> to the right or left on the x-axis, or up or down on the y-axis.<br><br> <img src="/images/domain-range-img1.png" alt="Domain range and relation from ordered pairs"> <br> <b>Relation</b> A relation describes how ordered pairs are related and, more specifically, whether or not they are functions. A relation between ordered pairs is a function if it has only one y-value (output) for each x-value (input). If an x-value has more than one y-value, then the relation is not a function.<br> <img src="/images/domain-range-img2.png" alt="Domain range and relation from ordered pairs"> <br> <b>Function</b> A function is a relation that takes a chosen input and creates an output. For example, the function <math>f(x)=2x</math> multiplies every input <math>(x)</math> by <math>2</math> to create an output. A relation between ordered pairs is a function if it has only one y-value (output) for each x-value (input). If an x-value has more than one y-value, then the relation is not a function. <br><br> <b>Domain</b> The domain represents all the possible inputs for a variable of a function. It is considered an independent variable because it represents the quantity being manipulated and is not dependent on any other factor. Its value is represented by the first number in an ordered pair. For example, in the input <math>(4, -5)</math> , <math>4</math> is the domain. <br><br> <b>Range</b> The range represents all the possible outputs of a function. It is considered a dependant variable because it is dependent on how the independent variable is manipulated. Its value is represented by the second number in an ordered pair. In the ordered pair <math>(4, -5)</math> , <math>-5</math> is the range. <br> <img src="/images/domain-range-img3.png" alt="Domain range and relation from ordered pairs">	Para poder averiguar el dominio y el rango de pares ordenados y determinar si son funciones, antes tenemos que entender unos cuantos conceptos clave.<br><br> <b>Par ordenado</b><br> Un par ordenado está representado por dos números que se escriben en un orden específico, normalmente entre paréntesis, con una coma entre ambos números. Por ejemplo: <math>(4, -5)</math> es un par ordenado. En un plano xy, también llamado plano cartesiano, un par ordenado representa la ubicación de un punto; el primer número del paréntesis es la coordenada x del punto y el segundo número la coordenada y. Para dibujar estas coordenadas en un plano, nos desplazamos hacia fuera desde el origen del plano <math>(0,0)</math>: hacia la derecha o hacia la izquierda en el eje x, y hacia arriba o hacia abajo en el eje y.<br><br> <img src="/images/domain-range-img1.png" alt="Domain range and relation from ordered pairs"> <br> <b>Relación</b> Una relación describe cómo están relacionados los pares ordenados y, más concretamente, si son o no funciones. Una relación entre pares ordenados únicamente es una función si tiene un solo valor de y (salida) para cada valor de x (entrada). Si un valor de x tiene más de un valor de y, entonces la relación no es una función.<br> <img src="/images/domain-range-img2.png" alt="Domain range and relation from ordered pairs"> <br> <b>Función</b> Una función es una relación que toma un valor de entrada elegido y crea un valor de salida. Por ejemplo, la función <math>f(x)=2x</math> multiplica cada entrada <math>(x)</math> por <math>2</math> para obtener el valor de salida. Una relación entre pares ordenados únicamente es una función si tiene un solo valor de y (salida) para cada valor de x (entrada). Si un valor de x tiene más de un valor de y, entonces la relación no es una función. <br><br> <b>Dominio</b> El dominio representa todas las posibles entradas para la variable de una función. Se considera una variable independiente porque representa la cantidad que se manipula y no depende de ningún otro factor. Su valor está representado por el primer número de un par ordenado. Por ejemplo, en el par ordenado <math>(4, -5)</math>, <math>4</math> es el dominio. <br><br> <b>Rango</b> El rango representa todas las posibles salidas de una función. Se considera una variable dependiente porque depende de cómo se manipule la variable independiente. Su valor está representado por el segundo número de un par ordenado. En el par ordenado <math>(4, -5)</math>, <math>-5</math> es el rango. <br> <img src="/images/domain-range-img3.png" alt="Domain range and relation from ordered pairs">
DomainRangeAndRelationFromOrderedPairsTopicTitle	Domain range and relation from ordered pairs	Dominio, rango y relación de pares ordenados
DomainRangeRelationFromOrderedPairs	Finding the domain and range of a relation from ordered pairs	Averiguar el dominio y el rango de una relación a partir de pares ordenados
DomainRangeRelationFromOrderedPairs3TableTitle	A relation is considered a function when every x-input has only one y-output.	Una relación se considera una función cuando cada entrada “x” tiene una sola salida “y”.
DomainRangeRelationFromOrderedPairsFinalResultIsFunction	The list of ordered pairs is a function.	La lista de pares ordenados es una función.
DomainRangeRelationFromOrderedPairsFinalResultIsNotFunction	The list of ordered pairs is not a function.	La lista de pares ordenados no es una función.
DomainRangeRelationFromOrderedPairsFinalResultTypeDomain	Domain:	Dominio:
DomainRangeRelationFromOrderedPairsFinalResultTypeRange	Range:	Rango:
DomainRangeRelationFromOrderedPairs.plugin	finding the domain and range of a relation from ordered pairs	determinar el dominio y el rango de una relación a partir de pares ordenados
DomainRangeRelationFromOrderedPairsStep1TextUnit1	The domain is a set made up of the x-values of the ordered pairs: {Step1BoldedListOfPoints}<br><br> Domain: {Step1DomainList}	El dominio es un conjunto formado por los valores x de los pares ordenados: {Step1BoldedListOfPoints}<br><br> Dominio: {Step1DomainList}