Solve (2)/(2(x)^(2)+5x+3)-(1)/(2(x)^(2)-x-6)+(3)/((x)^(2)-x-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

$$\frac{ 2 }{ 2 { x }^{ 2 } +5x+3 } - \frac{ 1 }{ 2 { x }^{ 2 } -x-6 } + \frac{ 3 }{ { x }^{ 2 } -x-2 }$$

Evaluate

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

Short Solution Steps

Factor $2x^{2}+5x+3$. Factor $2x^{2}-x-6$.

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

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

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

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

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

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

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

Combine like terms in $2x-4-x-1$.

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

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

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

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

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

Since $\frac{x-5}{\left(x-2\right)\left(x+1\right)\left(2x+3\right)}$ and $\frac{3\left(2x+3\right)}{\left(x-2\right)\left(x+1\right)\left(2x+3\right)}$ have the same denominator, add them by adding their numerators.

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

Do the multiplications in $x-5+3\left(2x+3\right)$.

$$\frac{x-5+6x+9}{\left(x-2\right)\left(x+1\right)\left(2x+3\right)}$$

Combine like terms in $x-5+6x+9$.

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

Expand $\left(x-2\right)\left(x+1\right)\left(2x+3\right)$.

$$\frac{7x+4}{2x^{3}+x^{2}-7x-6}$$

Differentiate w.r.t. x

$-\frac{28x^{3}+31x^{2}+8x+14}{\left(\left(x-2\right)\left(x+1\right)\left(2x+3\right)\right)^{2}}$