Solve (x ^ 2 + 12x + 36)/(x ^ 2 - 16) + (3x + 18)/(2x ^ 2 + 8x) | 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

$$\frac{x^{2}+12x+36}{x^{2}-16}+\frac{3x+18}{2x^{2}+8x}$$

Evaluate

$\frac{\left(x+6\right)\left(2x^{2}+15x-12\right)}{2x\left(x^{2}-16\right)}$

Short Solution Steps

Factor $x^{2}-16$. Factor $2x^{2}+8x$.

$$\frac{x^{2}+12x+36}{\left(x-4\right)\left(x+4\right)}+\frac{3x+18}{2x\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-4\right)\left(x+4\right)$ and $2x\left(x+4\right)$ is $2x\left(x-4\right)\left(x+4\right)$. Multiply $\frac{x^{2}+12x+36}{\left(x-4\right)\left(x+4\right)}$ times $\frac{2x}{2x}$. Multiply $\frac{3x+18}{2x\left(x+4\right)}$ times $\frac{x-4}{x-4}$.

$$\frac{\left(x^{2}+12x+36\right)\times 2x}{2x\left(x-4\right)\left(x+4\right)}+\frac{\left(3x+18\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Since $\frac{\left(x^{2}+12x+36\right)\times 2x}{2x\left(x-4\right)\left(x+4\right)}$ and $\frac{\left(3x+18\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}+12x+36\right)\times 2x+\left(3x+18\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in $\left(x^{2}+12x+36\right)\times 2x+\left(3x+18\right)\left(x-4\right)$.

$$\frac{2x^{3}+24x^{2}+72x+3x^{2}-12x+18x-72}{2x\left(x-4\right)\left(x+4\right)}$$

Combine like terms in $2x^{3}+24x^{2}+72x+3x^{2}-12x+18x-72$.

$$\frac{2x^{3}+27x^{2}+78x-72}{2x\left(x-4\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in $\frac{2x^{3}+27x^{2}+78x-72}{2x\left(x-4\right)\left(x+4\right)}$.

$$\frac{2\left(x+6\right)\left(x^{2}+\frac{15}{2}x-6\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{\left(x+6\right)\left(x^{2}+\frac{15}{2}x-6\right)}{x\left(x-4\right)\left(x+4\right)}$$

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

$$\frac{\left(x+6\right)\left(x^{2}+\frac{15}{2}x-6\right)}{x^{3}-16x}$$

Use the distributive property to multiply $x+6$ by $x^{2}+\frac{15}{2}x-6$ and combine like terms.

$$\frac{x^{3}+\frac{27}{2}x^{2}+39x-36}{x^{3}-16x}$$

Expand

$\frac{2x^{3}+27x^{2}+78x-72}{2x\left(x^{2}-16\right)}$

Short Solution Steps

Factor $x^{2}-16$. Factor $2x^{2}+8x$.

$$\frac{x^{2}+12x+36}{\left(x-4\right)\left(x+4\right)}+\frac{3x+18}{2x\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-4\right)\left(x+4\right)$ and $2x\left(x+4\right)$ is $2x\left(x-4\right)\left(x+4\right)$. Multiply $\frac{x^{2}+12x+36}{\left(x-4\right)\left(x+4\right)}$ times $\frac{2x}{2x}$. Multiply $\frac{3x+18}{2x\left(x+4\right)}$ times $\frac{x-4}{x-4}$.

$$\frac{\left(x^{2}+12x+36\right)\times 2x}{2x\left(x-4\right)\left(x+4\right)}+\frac{\left(3x+18\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Since $\frac{\left(x^{2}+12x+36\right)\times 2x}{2x\left(x-4\right)\left(x+4\right)}$ and $\frac{\left(3x+18\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x^{2}+12x+36\right)\times 2x+\left(3x+18\right)\left(x-4\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in $\left(x^{2}+12x+36\right)\times 2x+\left(3x+18\right)\left(x-4\right)$.

$$\frac{2x^{3}+24x^{2}+72x+3x^{2}-12x+18x-72}{2x\left(x-4\right)\left(x+4\right)}$$

Combine like terms in $2x^{3}+24x^{2}+72x+3x^{2}-12x+18x-72$.

$$\frac{2x^{3}+27x^{2}+78x-72}{2x\left(x-4\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in $\frac{2x^{3}+27x^{2}+78x-72}{2x\left(x-4\right)\left(x+4\right)}$.

$$\frac{2\left(x+6\right)\left(x^{2}+\frac{15}{2}x-6\right)}{2x\left(x-4\right)\left(x+4\right)}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{\left(x+6\right)\left(x^{2}+\frac{15}{2}x-6\right)}{x\left(x-4\right)\left(x+4\right)}$$

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

$$\frac{\left(x+6\right)\left(x^{2}+\frac{15}{2}x-6\right)}{x^{3}-16x}$$

Use the distributive property to multiply $x+6$ by $x^{2}+\frac{15}{2}x-6$ and combine like terms.

$$\frac{x^{3}+\frac{27}{2}x^{2}+39x-36}{x^{3}-16x}$$