Solve x/(x-7)^2+4/x^2-49

$$\frac { x } { ( x - 7 ) ^ { 2 } } + \frac { 4 } { x ^ { 2 } - 49 }$$

Evaluate

$$\frac{x^{2}+11x-28}{\left(x+7\right)\left(x-7\right)^{2}}$$

Short Solution Steps

Factor x^{2}-49.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)^{2} and \left(x-7\right)\left(x+7\right) is \left(x+7\right)\left(x-7\right)^{2}. Multiply \frac{x}{\left(x-7\right)^{2}} times \frac{x+7}{x+7}. Multiply \frac{4}{\left(x-7\right)\left(x+7\right)} times \frac{x-7}{x-7}.

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

Since \frac{x\left(x+7\right)}{\left(x+7\right)\left(x-7\right)^{2}} and \frac{4\left(x-7\right)}{\left(x+7\right)\left(x-7\right)^{2}} have the same denominator, add them by adding their numerators.

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

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

$$\frac{x^{2}+7x+4x-28}{\left(x+7\right)\left(x-7\right)^{2}}$$

Combine like terms in x^{2}+7x+4x-28.

$$\frac{x^{2}+11x-28}{\left(x+7\right)\left(x-7\right)^{2}}$$

Expand \left(x+7\right)\left(x-7\right)^{2}.

$$\frac{x^{2}+11x-28}{x^{3}-7x^{2}-49x+343}$$

Expand

$$\frac{x^{2}+11x-28}{\left(x+7\right)\left(x-7\right)^{2}}$$

Short Solution Steps

Factor x^{2}-49.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-7\right)^{2} and \left(x-7\right)\left(x+7\right) is \left(x+7\right)\left(x-7\right)^{2}. Multiply \frac{x}{\left(x-7\right)^{2}} times \frac{x+7}{x+7}. Multiply \frac{4}{\left(x-7\right)\left(x+7\right)} times \frac{x-7}{x-7}.

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

Since \frac{x\left(x+7\right)}{\left(x+7\right)\left(x-7\right)^{2}} and \frac{4\left(x-7\right)}{\left(x+7\right)\left(x-7\right)^{2}} have the same denominator, add them by adding their numerators.

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

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

$$\frac{x^{2}+7x+4x-28}{\left(x+7\right)\left(x-7\right)^{2}}$$

Combine like terms in x^{2}+7x+4x-28.

$$\frac{x^{2}+11x-28}{\left(x+7\right)\left(x-7\right)^{2}}$$

Expand \left(x+7\right)\left(x-7\right)^{2}.

$$\frac{x^{2}+11x-28}{x^{3}-7x^{2}-49x+343}$$