Solve - 3/(x ^ 2 + 2x - 8) 2 x ^ 2 + 5x + 4 -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

$$\frac { - 3 } { x ^ { 2 } + 2 x - 8 } + \frac { 2 } { x ^ { 2 } + 5 x + 4 } - \frac { 1 } { x - x - 2 }$$

Evaluate

$\frac{x^{3}+3x^{2}-8x-22}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}$

Short Solution Steps

Combine $x$ and $-x$ to get $0$.

$$\frac{-3}{x^{2}+2x-8}+\frac{2}{x^{2}+5x+4}-\frac{1}{-2}$$

Fraction $\frac{1}{-2}$ can be rewritten as $-\frac{1}{2}$ by extracting the negative sign.

$$\frac{-3}{x^{2}+2x-8}+\frac{2}{x^{2}+5x+4}-\left(-\frac{1}{2}\right)$$

The opposite of $-\frac{1}{2}$ is $\frac{1}{2}$.

$$\frac{-3}{x^{2}+2x-8}+\frac{2}{x^{2}+5x+4}+\frac{1}{2}$$

Factor $x^{2}+2x-8$. Factor $x^{2}+5x+4$.

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

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

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

Since $\frac{-3\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ and $\frac{2\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $-3\left(x+1\right)+2\left(x-2\right)$.

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

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

$$\frac{-x-7}{\left(x-2\right)\left(x+1\right)\left(x+4\right)}+\frac{1}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-2\right)\left(x+1\right)\left(x+4\right)$ and $2$ is $2\left(x-2\right)\left(x+1\right)\left(x+4\right)$. Multiply $\frac{-x-7}{\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ times $\frac{2}{2}$. Multiply $\frac{1}{2}$ times $\frac{\left(x-2\right)\left(x+1\right)\left(x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+4\right)}$.

$$\frac{2\left(-x-7\right)}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}+\frac{\left(x-2\right)\left(x+1\right)\left(x+4\right)}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Since $\frac{2\left(-x-7\right)}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ and $\frac{\left(x-2\right)\left(x+1\right)\left(x+4\right)}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{2\left(-x-7\right)+\left(x-2\right)\left(x+1\right)\left(x+4\right)}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Do the multiplications in $2\left(-x-7\right)+\left(x-2\right)\left(x+1\right)\left(x+4\right)$.

$$\frac{-2x-14+x^{3}+5x^{2}+4x-2x^{2}-10x-8}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Combine like terms in $-2x-14+x^{3}+5x^{2}+4x-2x^{2}-10x-8$.

$$\frac{-8x-22+x^{3}+3x^{2}}{2\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Expand $2\left(x-2\right)\left(x+1\right)\left(x+4\right)$.

$$\frac{-8x-22+x^{3}+3x^{2}}{2x^{3}+6x^{2}-12x-16}$$

Differentiate w.r.t. x

$\frac{2\left(x^{3}+12x^{2}+21x-17\right)}{\left(\left(x-2\right)\left(x+1\right)\left(x+4\right)\right)^{2}}$