Solve (3/(x ^ 2 - 4) - 2/(x ^ 2 - 4x + 4)) / ((2x - 20)/(3x ^ 2 - 12)) | 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

$$(\frac{3}{x^{2}-4}-\frac{2}{x^{2}-4x+4})\div\frac{2x-20}{3x^{2}-12}$$

Evaluate

$\frac{3}{2\left(x-2\right)}$

Short Solution Steps

Factor $x^{2}-4$. Factor $x^{2}-4x+4$.

$$\frac{\frac{3}{\left(x-2\right)\left(x+2\right)}-\frac{2}{\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-2\right)\left(x+2\right)$ and $\left(x-2\right)^{2}$ is $\left(x+2\right)\left(x-2\right)^{2}$. Multiply $\frac{3}{\left(x-2\right)\left(x+2\right)}$ times $\frac{x-2}{x-2}$. Multiply $\frac{2}{\left(x-2\right)^{2}}$ times $\frac{x+2}{x+2}$.

$$\frac{\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}-\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Since $\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}$ and $\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3\left(x-2\right)-2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Do the multiplications in $3\left(x-2\right)-2\left(x+2\right)$.

$$\frac{\frac{3x-6-2x-4}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Combine like terms in $3x-6-2x-4$.

$$\frac{\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Divide $\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}$ by $\frac{2x-20}{3x^{2}-12}$ by multiplying $\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}$ by the reciprocal of $\frac{2x-20}{3x^{2}-12}$.

$$\frac{\left(x-10\right)\left(3x^{2}-12\right)}{\left(x+2\right)\left(x-2\right)^{2}\left(2x-20\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(x-10\right)\left(x-2\right)\left(x+2\right)}{2\left(x-10\right)\left(x+2\right)\left(x-2\right)^{2}}$$

Cancel out $\left(x-10\right)\left(x-2\right)\left(x+2\right)$ in both numerator and denominator.

$$\frac{3}{2\left(x-2\right)}$$

Expand the expression.

$$\frac{3}{2x-4}$$

Expand

$\frac{3}{2\left(x-2\right)}$

Short Solution Steps

Factor $x^{2}-4$. Factor $x^{2}-4x+4$.

$$\frac{\frac{3}{\left(x-2\right)\left(x+2\right)}-\frac{2}{\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-2\right)\left(x+2\right)$ and $\left(x-2\right)^{2}$ is $\left(x+2\right)\left(x-2\right)^{2}$. Multiply $\frac{3}{\left(x-2\right)\left(x+2\right)}$ times $\frac{x-2}{x-2}$. Multiply $\frac{2}{\left(x-2\right)^{2}}$ times $\frac{x+2}{x+2}$.

$$\frac{\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}-\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Since $\frac{3\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}}$ and $\frac{2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3\left(x-2\right)-2\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Do the multiplications in $3\left(x-2\right)-2\left(x+2\right)$.

$$\frac{\frac{3x-6-2x-4}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Combine like terms in $3x-6-2x-4$.

$$\frac{\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}}{\frac{2x-20}{3x^{2}-12}}$$

Divide $\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}$ by $\frac{2x-20}{3x^{2}-12}$ by multiplying $\frac{x-10}{\left(x+2\right)\left(x-2\right)^{2}}$ by the reciprocal of $\frac{2x-20}{3x^{2}-12}$.

$$\frac{\left(x-10\right)\left(3x^{2}-12\right)}{\left(x+2\right)\left(x-2\right)^{2}\left(2x-20\right)}$$

Factor the expressions that are not already factored.

$$\frac{3\left(x-10\right)\left(x-2\right)\left(x+2\right)}{2\left(x-10\right)\left(x+2\right)\left(x-2\right)^{2}}$$

Cancel out $\left(x-10\right)\left(x-2\right)\left(x+2\right)$ in both numerator and denominator.

$$\frac{3}{2\left(x-2\right)}$$

Expand the expression.

$$\frac{3}{2x-4}$$