Solve x^2+x/x^2-49+x^2-1/x^2+17x+70

$$\frac { x ^ { 2 } + x } { x ^ { 2 } - 49 } + \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 17 x + 70 }$$

Evaluate

$$\frac{\left(x+1\right)\left(2x^{2}+2x+7\right)}{\left(x+10\right)\left(x^{2}-49\right)}$$

Short Solution Steps

Factor x^{2}-49. Factor x^{2}+17x+70.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x+7\right) and \left(x+7\right)\left(x+10\right) is \left(x-7\right)\left(x+7\right)\left(x+10\right). Multiply \frac{x^{2}+x}{\left(x-7\right)\left(x+7\right)} times \frac{x+10}{x+10}. Multiply \frac{x^{2}-1}{\left(x+7\right)\left(x+10\right)} times \frac{x-7}{x-7}.

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

Since \frac{\left(x^{2}+x\right)\left(x+10\right)}{\left(x-7\right)\left(x+7\right)\left(x+10\right)} and \frac{\left(x^{2}-1\right)\left(x-7\right)}{\left(x-7\right)\left(x+7\right)\left(x+10\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}+x\right)\left(x+10\right)+\left(x^{2}-1\right)\left(x-7\right).

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

Combine like terms in x^{3}+10x^{2}+x^{2}+10x+x^{3}-7x^{2}-x+7.

$$\frac{2x^{3}+4x^{2}+9x+7}{\left(x-7\right)\left(x+7\right)\left(x+10\right)}$$

Expand \left(x-7\right)\left(x+7\right)\left(x+10\right).

$$\frac{2x^{3}+4x^{2}+9x+7}{x^{3}+10x^{2}-49x-490}$$

Expand

$$\frac{2x^{3}+4x^{2}+9x+7}{\left(x+10\right)\left(x^{2}-49\right)}$$

Short Solution Steps

Factor x^{2}-49. Factor x^{2}+17x+70.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)\left(x+7\right) and \left(x+7\right)\left(x+10\right) is \left(x-7\right)\left(x+7\right)\left(x+10\right). Multiply \frac{x^{2}+x}{\left(x-7\right)\left(x+7\right)} times \frac{x+10}{x+10}. Multiply \frac{x^{2}-1}{\left(x+7\right)\left(x+10\right)} times \frac{x-7}{x-7}.

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

Since \frac{\left(x^{2}+x\right)\left(x+10\right)}{\left(x-7\right)\left(x+7\right)\left(x+10\right)} and \frac{\left(x^{2}-1\right)\left(x-7\right)}{\left(x-7\right)\left(x+7\right)\left(x+10\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(x^{2}+x\right)\left(x+10\right)+\left(x^{2}-1\right)\left(x-7\right).

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

Combine like terms in x^{3}+10x^{2}+x^{2}+10x+x^{3}-7x^{2}-x+7.

$$\frac{2x^{3}+4x^{2}+9x+7}{\left(x-7\right)\left(x+7\right)\left(x+10\right)}$$

Expand \left(x-7\right)\left(x+7\right)\left(x+10\right).

$$\frac{2x^{3}+4x^{2}+9x+7}{x^{3}+10x^{2}-49x-490}$$