Solve x ^ 2 - y ^ 2 x x ^ 3 + y ^ 3; (x-y)^ 2 +xy/ (x+2y)^ 2 -2xy x^ 3 +xy^ 2 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac { x ^ { 2 } - y ^ { 2 } } { x ^ { 3 } + y ^ { 3 } } \times \frac { ( x - y ) ^ { 2 } + x y } { x ^ { 2 } - x y } \div \frac { ( x + 2 y ) ^ { 2 } - 2 x y } { x ^ { 3 } + x y ^ { 2 } }$$

Evaluate

$\frac{x^{2}+y^{2}}{x^{2}+2xy+4y^{2}}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{x^{2}-y^{2}}{x^{3}+y^{3}}$.

$$\frac{\frac{\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}\times \frac{\left(x-y\right)^{2}+xy}{x^{2}-xy}}{\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}}$$

Cancel out $x+y$ in both numerator and denominator.

$$\frac{\frac{x-y}{x^{2}-xy+y^{2}}\times \frac{\left(x-y\right)^{2}+xy}{x^{2}-xy}}{\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}}$$

Multiply $\frac{x-y}{x^{2}-xy+y^{2}}$ times $\frac{\left(x-y\right)^{2}+xy}{x^{2}-xy}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)}}{\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}}$$

Divide $\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)}$ by $\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}$ by multiplying $\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)}$ by the reciprocal of $\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}$.

$$\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)\left(x^{3}+xy^{2}\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)\left(\left(x+2y\right)^{2}-2xy\right)}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x-y\right)\left(x^{2}+y^{2}\right)\left(x^{2}-xy+y^{2}\right)}{x\left(x-y\right)\left(x^{2}-xy+y^{2}\right)\left(x^{2}+2xy+4y^{2}\right)}$$

Cancel out $x\left(x-y\right)\left(x^{2}-xy+y^{2}\right)$ in both numerator and denominator.

$$\frac{x^{2}+y^{2}}{x^{2}+2xy+4y^{2}}$$

Expand

$\frac{x^{2}+y^{2}}{x^{2}+2xy+4y^{2}}$

Short Solution Steps

Factor the expressions that are not already factored in $\frac{x^{2}-y^{2}}{x^{3}+y^{3}}$.

$$\frac{\frac{\left(x+y\right)\left(x-y\right)}{\left(x+y\right)\left(x^{2}-xy+y^{2}\right)}\times \frac{\left(x-y\right)^{2}+xy}{x^{2}-xy}}{\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}}$$

Cancel out $x+y$ in both numerator and denominator.

$$\frac{\frac{x-y}{x^{2}-xy+y^{2}}\times \frac{\left(x-y\right)^{2}+xy}{x^{2}-xy}}{\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}}$$

Multiply $\frac{x-y}{x^{2}-xy+y^{2}}$ times $\frac{\left(x-y\right)^{2}+xy}{x^{2}-xy}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)}}{\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}}$$

Divide $\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)}$ by $\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}$ by multiplying $\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)}$ by the reciprocal of $\frac{\left(x+2y\right)^{2}-2xy}{x^{3}+xy^{2}}$.

$$\frac{\left(x-y\right)\left(\left(x-y\right)^{2}+xy\right)\left(x^{3}+xy^{2}\right)}{\left(x^{2}-xy+y^{2}\right)\left(x^{2}-xy\right)\left(\left(x+2y\right)^{2}-2xy\right)}$$

Factor the expressions that are not already factored.

$$\frac{x\left(x-y\right)\left(x^{2}+y^{2}\right)\left(x^{2}-xy+y^{2}\right)}{x\left(x-y\right)\left(x^{2}-xy+y^{2}\right)\left(x^{2}+2xy+4y^{2}\right)}$$

Cancel out $x\left(x-y\right)\left(x^{2}-xy+y^{2}\right)$ in both numerator and denominator.

$$\frac{x^{2}+y^{2}}{x^{2}+2xy+4y^{2}}$$