Solve (3x)/(2(x)^(2)-2x-4)-(x-1)/(4(x)^(2)+8x-32)-(10x-1)/(8(x)^(2)+40x+32) | 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

$$\frac{ 3x }{ 2 { x }^{ 2 } -2x-4 } - \frac{ x-1 }{ 4 { x }^{ 2 } +8x-32 } - \frac{ 10x-1 }{ 8 { x }^{ 2 } +40x+32 }$$

Evaluate

$\frac{69x}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$

Short Solution Steps

Factor $2x^{2}-2x-4$. Factor $4x^{2}+8x-32$.

$$\frac{3x}{2\left(x-2\right)\left(x+1\right)}-\frac{x-1}{4\left(x-2\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

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

$$\frac{3x\times 2\left(x+4\right)}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{\left(x-1\right)\left(x+1\right)}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

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

$$\frac{3x\times 2\left(x+4\right)-\left(x-1\right)\left(x+1\right)}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

Do the multiplications in $3x\times 2\left(x+4\right)-\left(x-1\right)\left(x+1\right)$.

$$\frac{6x^{2}+24x-x^{2}-x+x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

Combine like terms in $6x^{2}+24x-x^{2}-x+x+1$.

$$\frac{5x^{2}+24x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

Factor $8x^{2}+40x+32$.

$$\frac{5x^{2}+24x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8\left(x+1\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4\left(x-2\right)\left(x+1\right)\left(x+4\right)$ and $8\left(x+1\right)\left(x+4\right)$ is $8\left(x-2\right)\left(x+1\right)\left(x+4\right)$. Multiply $\frac{5x^{2}+24x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ times $\frac{2}{2}$. Multiply $\frac{10x-1}{8\left(x+1\right)\left(x+4\right)}$ times $\frac{x-2}{x-2}$.

$$\frac{2\left(5x^{2}+24x+1\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{\left(10x-1\right)\left(x-2\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Since $\frac{2\left(5x^{2}+24x+1\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ and $\frac{\left(10x-1\right)\left(x-2\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(5x^{2}+24x+1\right)-\left(10x-1\right)\left(x-2\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Do the multiplications in $2\left(5x^{2}+24x+1\right)-\left(10x-1\right)\left(x-2\right)$.

$$\frac{10x^{2}+48x+2-10x^{2}+20x+x-2}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Combine like terms in $10x^{2}+48x+2-10x^{2}+20x+x-2$.

$$\frac{69x}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Expand $8\left(x-2\right)\left(x+1\right)\left(x+4\right)$.

$$\frac{69x}{8x^{3}+24x^{2}-48x-64}$$

Expand

$\frac{69x}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$

Short Solution Steps

Factor $2x^{2}-2x-4$. Factor $4x^{2}+8x-32$.

$$\frac{3x}{2\left(x-2\right)\left(x+1\right)}-\frac{x-1}{4\left(x-2\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

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

$$\frac{3x\times 2\left(x+4\right)}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{\left(x-1\right)\left(x+1\right)}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

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

$$\frac{3x\times 2\left(x+4\right)-\left(x-1\right)\left(x+1\right)}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

Do the multiplications in $3x\times 2\left(x+4\right)-\left(x-1\right)\left(x+1\right)$.

$$\frac{6x^{2}+24x-x^{2}-x+x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

Combine like terms in $6x^{2}+24x-x^{2}-x+x+1$.

$$\frac{5x^{2}+24x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8x^{2}+40x+32}$$

Factor $8x^{2}+40x+32$.

$$\frac{5x^{2}+24x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{10x-1}{8\left(x+1\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $4\left(x-2\right)\left(x+1\right)\left(x+4\right)$ and $8\left(x+1\right)\left(x+4\right)$ is $8\left(x-2\right)\left(x+1\right)\left(x+4\right)$. Multiply $\frac{5x^{2}+24x+1}{4\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ times $\frac{2}{2}$. Multiply $\frac{10x-1}{8\left(x+1\right)\left(x+4\right)}$ times $\frac{x-2}{x-2}$.

$$\frac{2\left(5x^{2}+24x+1\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}-\frac{\left(10x-1\right)\left(x-2\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Since $\frac{2\left(5x^{2}+24x+1\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ and $\frac{\left(10x-1\right)\left(x-2\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$ have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\left(5x^{2}+24x+1\right)-\left(10x-1\right)\left(x-2\right)}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Do the multiplications in $2\left(5x^{2}+24x+1\right)-\left(10x-1\right)\left(x-2\right)$.

$$\frac{10x^{2}+48x+2-10x^{2}+20x+x-2}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Combine like terms in $10x^{2}+48x+2-10x^{2}+20x+x-2$.

$$\frac{69x}{8\left(x-2\right)\left(x+1\right)\left(x+4\right)}$$

Expand $8\left(x-2\right)\left(x+1\right)\left(x+4\right)$.

$$\frac{69x}{8x^{3}+24x^{2}-48x-64}$$