Solve (a + b)/(a - b) - (a - b)/(a + b) - (2ab)/(a ^ 2 - b ^ 2) | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$\frac{a+b}{a-b}-\frac{a-b}{a+b}-\frac{2ab}{a^{2}-b^{2}}$$

Evaluate

$\frac{2ab}{a^{2}-b^{2}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a-b$ and $a+b$ is $\left(a+b\right)\left(a-b\right)$. Multiply $\frac{a+b}{a-b}$ times $\frac{a+b}{a+b}$. Multiply $\frac{a-b}{a+b}$ times $\frac{a-b}{a-b}$.

$$\frac{\left(a+b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{\left(a-b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Since $\frac{\left(a+b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}$ and $\frac{\left(a-b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+b\right)\left(a+b\right)-\left(a-b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Do the multiplications in $\left(a+b\right)\left(a+b\right)-\left(a-b\right)\left(a-b\right)$.

$$\frac{a^{2}+ab+ab+b^{2}-a^{2}+ab+ab-b^{2}}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Combine like terms in $a^{2}+ab+ab+b^{2}-a^{2}+ab+ab-b^{2}$.

$$\frac{4ab}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Factor $a^{2}-b^{2}$.

$$\frac{4ab}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{\left(a+b\right)\left(a-b\right)}$$

Since $\frac{4ab}{\left(a+b\right)\left(a-b\right)}$ and $\frac{2ab}{\left(a+b\right)\left(a-b\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{4ab-2ab}{\left(a+b\right)\left(a-b\right)}$$

Combine like terms in $4ab-2ab$.

$$\frac{2ab}{\left(a+b\right)\left(a-b\right)}$$

Expand $\left(a+b\right)\left(a-b\right)$.

$$\frac{2ab}{a^{2}-b^{2}}$$

Expand

$\frac{2ab}{a^{2}-b^{2}}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $a-b$ and $a+b$ is $\left(a+b\right)\left(a-b\right)$. Multiply $\frac{a+b}{a-b}$ times $\frac{a+b}{a+b}$. Multiply $\frac{a-b}{a+b}$ times $\frac{a-b}{a-b}$.

$$\frac{\left(a+b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{\left(a-b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Since $\frac{\left(a+b\right)\left(a+b\right)}{\left(a+b\right)\left(a-b\right)}$ and $\frac{\left(a-b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(a+b\right)\left(a+b\right)-\left(a-b\right)\left(a-b\right)}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Do the multiplications in $\left(a+b\right)\left(a+b\right)-\left(a-b\right)\left(a-b\right)$.

$$\frac{a^{2}+ab+ab+b^{2}-a^{2}+ab+ab-b^{2}}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Combine like terms in $a^{2}+ab+ab+b^{2}-a^{2}+ab+ab-b^{2}$.

$$\frac{4ab}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{a^{2}-b^{2}}$$

Factor $a^{2}-b^{2}$.

$$\frac{4ab}{\left(a+b\right)\left(a-b\right)}-\frac{2ab}{\left(a+b\right)\left(a-b\right)}$$

Since $\frac{4ab}{\left(a+b\right)\left(a-b\right)}$ and $\frac{2ab}{\left(a+b\right)\left(a-b\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{4ab-2ab}{\left(a+b\right)\left(a-b\right)}$$

Combine like terms in $4ab-2ab$.

$$\frac{2ab}{\left(a+b\right)\left(a-b\right)}$$

Expand $\left(a+b\right)\left(a-b\right)$.

$$\frac{2ab}{a^{2}-b^{2}}$$