Solve -3 + 2x - 8 + 2 + 4 1 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

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

Evaluate

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

Short Solution Steps

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

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

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 $\frac{x-2}{x-2}$.

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

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.

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

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

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

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

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

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

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

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

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

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

$$\frac{-x^{2}-7x-x^{3}-5x^{2}-4x+2x^{2}+10x+8}{x\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Combine like terms in $-x^{2}-7x-x^{3}-5x^{2}-4x+2x^{2}+10x+8$.

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

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

$$\frac{-4x^{2}-x-x^{3}+8}{x^{4}+3x^{3}-6x^{2}-8x}$$

Differentiate w.r.t. x

$\frac{x^{6}+8x^{5}+21x^{4}-10x^{3}-46x^{2}+96x+64}{\left(x\left(x-2\right)\left(x+1\right)\left(x+4\right)\right)^{2}}$