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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$3/(x-3x-4)-x+1/4-x$$

Evaluate

$-\frac{8x^{2}+15x+4}{4\left(x+2\right)}$

Short Solution Steps

Combine $x$ and $-3x$ to get $-2x$.

$$\frac{3}{-2x-4}-x+\frac{1}{4}-x$$

Combine $-x$ and $-x$ to get $-2x$.

$$\frac{3}{-2x-4}-2x+\frac{1}{4}$$

Factor $-2x-4$.

$$\frac{3}{2\left(-x-2\right)}-2x+\frac{1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $2\left(-x-2\right)$ and $4$ is $4\left(x+2\right)$. Multiply $\frac{3}{2\left(-x-2\right)}$ times $\frac{-2}{-2}$. Multiply $\frac{1}{4}$ times $\frac{x+2}{x+2}$.

$$\frac{3\left(-2\right)}{4\left(x+2\right)}-2x+\frac{x+2}{4\left(x+2\right)}$$

Since $\frac{3\left(-2\right)}{4\left(x+2\right)}$ and $\frac{x+2}{4\left(x+2\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{3\left(-2\right)+x+2}{4\left(x+2\right)}-2x$$

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

$$\frac{-6+x+2}{4\left(x+2\right)}-2x$$

Combine like terms in $-6+x+2$.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply $-2x$ times $\frac{4\left(x+2\right)}{4\left(x+2\right)}$.

$$\frac{-4+x}{4\left(x+2\right)}+\frac{-2x\times 4\left(x+2\right)}{4\left(x+2\right)}$$

Since $\frac{-4+x}{4\left(x+2\right)}$ and $\frac{-2x\times 4\left(x+2\right)}{4\left(x+2\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-4+x-2x\times 4\left(x+2\right)}{4\left(x+2\right)}$$

Do the multiplications in $-4+x-2x\times 4\left(x+2\right)$.

$$\frac{-4+x-8x^{2}-16x}{4\left(x+2\right)}$$

Combine like terms in $-4+x-8x^{2}-16x$.

$$\frac{-4-15x-8x^{2}}{4\left(x+2\right)}$$

Factor the expressions that are not already factored in $\frac{-4-15x-8x^{2}}{4\left(x+2\right)}$.

$$\frac{-8\left(x-\left(-\frac{1}{16}\sqrt{97}-\frac{15}{16}\right)\right)\left(x-\left(\frac{1}{16}\sqrt{97}-\frac{15}{16}\right)\right)}{4\left(x+2\right)}$$

Cancel out $4$ in both numerator and denominator.

$$\frac{-2\left(x-\left(-\frac{1}{16}\sqrt{97}-\frac{15}{16}\right)\right)\left(x-\left(\frac{1}{16}\sqrt{97}-\frac{15}{16}\right)\right)}{x+2}$$

To find the opposite of $-\frac{1}{16}\sqrt{97}-\frac{15}{16}$, find the opposite of each term.

$$\frac{-2\left(x-\left(-\frac{1}{16}\sqrt{97}\right)-\left(-\frac{15}{16}\right)\right)\left(x-\left(\frac{1}{16}\sqrt{97}-\frac{15}{16}\right)\right)}{x+2}$$

The opposite of $-\frac{1}{16}\sqrt{97}$ is $\frac{1}{16}\sqrt{97}$.

$$\frac{-2\left(x+\frac{1}{16}\sqrt{97}-\left(-\frac{15}{16}\right)\right)\left(x-\left(\frac{1}{16}\sqrt{97}-\frac{15}{16}\right)\right)}{x+2}$$

The opposite of $-\frac{15}{16}$ is $\frac{15}{16}$.

$$\frac{-2\left(x+\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)\left(x-\left(\frac{1}{16}\sqrt{97}-\frac{15}{16}\right)\right)}{x+2}$$

To find the opposite of $\frac{1}{16}\sqrt{97}-\frac{15}{16}$, find the opposite of each term.

$$\frac{-2\left(x+\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)\left(x-\frac{1}{16}\sqrt{97}-\left(-\frac{15}{16}\right)\right)}{x+2}$$

The opposite of $-\frac{15}{16}$ is $\frac{15}{16}$.

$$\frac{-2\left(x+\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)\left(x-\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)}{x+2}$$

Use the distributive property to multiply $-2$ by $x+\frac{1}{16}\sqrt{97}+\frac{15}{16}$.

$$\frac{\left(-2x-2\times \frac{1}{16}\sqrt{97}-2\times \frac{15}{16}\right)\left(x-\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)}{x+2}$$

Multiply $-2$ and $\frac{1}{16}$ to get $\frac{-2}{16}$.

$$\frac{\left(-2x+\frac{-2}{16}\sqrt{97}-2\times \frac{15}{16}\right)\left(x-\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)}{x+2}$$

Reduce the fraction $\frac{-2}{16}$ to lowest terms by extracting and canceling out $2$.

$$\frac{\left(-2x-\frac{1}{8}\sqrt{97}-2\times \frac{15}{16}\right)\left(x-\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)}{x+2}$$

Express $-2\times \frac{15}{16}$ as a single fraction.

$$\frac{\left(-2x-\frac{1}{8}\sqrt{97}+\frac{-2\times 15}{16}\right)\left(x-\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)}{x+2}$$

Multiply $-2$ and $15$ to get $-30$.

$$\frac{\left(-2x-\frac{1}{8}\sqrt{97}+\frac{-30}{16}\right)\left(x-\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)}{x+2}$$

Reduce the fraction $\frac{-30}{16}$ to lowest terms by extracting and canceling out $2$.

$$\frac{\left(-2x-\frac{1}{8}\sqrt{97}-\frac{15}{8}\right)\left(x-\frac{1}{16}\sqrt{97}+\frac{15}{16}\right)}{x+2}$$

Apply the distributive property by multiplying each term of $-2x-\frac{1}{8}\sqrt{97}-\frac{15}{8}$ by each term of $x-\frac{1}{16}\sqrt{97}+\frac{15}{16}$.

$$\frac{-2x^{2}-2x\left(-\frac{1}{16}\right)\sqrt{97}-2x\times \frac{15}{16}-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\sqrt{97}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Multiply $\sqrt{97}$ and $\sqrt{97}$ to get $97$.

$$\frac{-2x^{2}-2x\left(-\frac{1}{16}\right)\sqrt{97}-2x\times \frac{15}{16}-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Express $-2\left(-\frac{1}{16}\right)$ as a single fraction.

$$\frac{-2x^{2}+\frac{-2\left(-1\right)}{16}x\sqrt{97}-2x\times \frac{15}{16}-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Multiply $-2$ and $-1$ to get $2$.

$$\frac{-2x^{2}+\frac{2}{16}x\sqrt{97}-2x\times \frac{15}{16}-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Reduce the fraction $\frac{2}{16}$ to lowest terms by extracting and canceling out $2$.

$$\frac{-2x^{2}+\frac{1}{8}x\sqrt{97}-2x\times \frac{15}{16}-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Express $-2\times \frac{15}{16}$ as a single fraction.

$$\frac{-2x^{2}+\frac{1}{8}x\sqrt{97}+\frac{-2\times 15}{16}x-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Multiply $-2$ and $15$ to get $-30$.

$$\frac{-2x^{2}+\frac{1}{8}x\sqrt{97}+\frac{-30}{16}x-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Reduce the fraction $\frac{-30}{16}$ to lowest terms by extracting and canceling out $2$.

$$\frac{-2x^{2}+\frac{1}{8}x\sqrt{97}-\frac{15}{8}x-\frac{1}{8}\sqrt{97}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Combine $\frac{1}{8}x\sqrt{97}$ and $-\frac{1}{8}\sqrt{97}x$ to get $0$.

$$\frac{-2x^{2}-\frac{15}{8}x-\frac{1}{8}\times 97\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Express $-\frac{1}{8}\times 97$ as a single fraction.

$$\frac{-2x^{2}-\frac{15}{8}x+\frac{-97}{8}\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Fraction $\frac{-97}{8}$ can be rewritten as $-\frac{97}{8}$ by extracting the negative sign.

$$\frac{-2x^{2}-\frac{15}{8}x-\frac{97}{8}\left(-\frac{1}{16}\right)-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Multiply $-\frac{97}{8}$ times $-\frac{1}{16}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}-\frac{15}{8}x+\frac{-97\left(-1\right)}{8\times 16}-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Do the multiplications in the fraction $\frac{-97\left(-1\right)}{8\times 16}$.

$$\frac{-2x^{2}-\frac{15}{8}x+\frac{97}{128}-\frac{1}{8}\sqrt{97}\times \frac{15}{16}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Multiply $-\frac{1}{8}$ times $\frac{15}{16}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}-\frac{15}{8}x+\frac{97}{128}+\frac{-15}{8\times 16}\sqrt{97}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Do the multiplications in the fraction $\frac{-15}{8\times 16}$.

$$\frac{-2x^{2}-\frac{15}{8}x+\frac{97}{128}+\frac{-15}{128}\sqrt{97}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Fraction $\frac{-15}{128}$ can be rewritten as $-\frac{15}{128}$ by extracting the negative sign.

$$\frac{-2x^{2}-\frac{15}{8}x+\frac{97}{128}-\frac{15}{128}\sqrt{97}-\frac{15}{8}x-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Combine $-\frac{15}{8}x$ and $-\frac{15}{8}x$ to get $-\frac{15}{4}x$.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97}{128}-\frac{15}{128}\sqrt{97}-\frac{15}{8}\left(-\frac{1}{16}\right)\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Multiply $-\frac{15}{8}$ times $-\frac{1}{16}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97}{128}-\frac{15}{128}\sqrt{97}+\frac{-15\left(-1\right)}{8\times 16}\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Do the multiplications in the fraction $\frac{-15\left(-1\right)}{8\times 16}$.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97}{128}-\frac{15}{128}\sqrt{97}+\frac{15}{128}\sqrt{97}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Combine $-\frac{15}{128}\sqrt{97}$ and $\frac{15}{128}\sqrt{97}$ to get $0$.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97}{128}-\frac{15}{8}\times \frac{15}{16}}{x+2}$$

Multiply $-\frac{15}{8}$ times $\frac{15}{16}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97}{128}+\frac{-15\times 15}{8\times 16}}{x+2}$$

Do the multiplications in the fraction $\frac{-15\times 15}{8\times 16}$.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97}{128}+\frac{-225}{128}}{x+2}$$

Fraction $\frac{-225}{128}$ can be rewritten as $-\frac{225}{128}$ by extracting the negative sign.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97}{128}-\frac{225}{128}}{x+2}$$

Since $\frac{97}{128}$ and $\frac{225}{128}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{97-225}{128}}{x+2}$$

Subtract $225$ from $97$ to get $-128$.

$$\frac{-2x^{2}-\frac{15}{4}x+\frac{-128}{128}}{x+2}$$

Divide $-128$ by $128$ to get $-1$.

$$\frac{-2x^{2}-\frac{15}{4}x-1}{x+2}$$

Factor

$-\frac{8\left(x-\frac{-\sqrt{97}-15}{16}\right)\left(x-\frac{\sqrt{97}-15}{16}\right)}{4\left(x+2\right)}$