Solve 4/x-2x/x^2-49+3/x+7=

$$\frac { 4 } { x } - \frac { 2 x } { x ^ { 2 } - 49 } + \frac { 3 } { x + 7 } =$$

Evaluate

$$\frac{5x^{2}-21x-196}{x\left(x^{2}-49\right)}$$

Short Solution Steps

Factor x^{2}-49.

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

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

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

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

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

Do the multiplications in 4\left(x-7\right)\left(x+7\right)-2xx.

$$\frac{4x^{2}+28x-28x-196-2x^{2}}{x\left(x-7\right)\left(x+7\right)}+\frac{3}{x+7}$$

Combine like terms in 4x^{2}+28x-28x-196-2x^{2}.

$$\frac{2x^{2}-196}{x\left(x-7\right)\left(x+7\right)}+\frac{3}{x+7}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-7\right)\left(x+7\right) and x+7 is x\left(x-7\right)\left(x+7\right). Multiply \frac{3}{x+7} times \frac{x\left(x-7\right)}{x\left(x-7\right)}.

$$\frac{2x^{2}-196}{x\left(x-7\right)\left(x+7\right)}+\frac{3x\left(x-7\right)}{x\left(x-7\right)\left(x+7\right)}$$

Since \frac{2x^{2}-196}{x\left(x-7\right)\left(x+7\right)} and \frac{3x\left(x-7\right)}{x\left(x-7\right)\left(x+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{2x^{2}-196+3x\left(x-7\right)}{x\left(x-7\right)\left(x+7\right)}$$

Do the multiplications in 2x^{2}-196+3x\left(x-7\right).

$$\frac{2x^{2}-196+3x^{2}-21x}{x\left(x-7\right)\left(x+7\right)}$$

Combine like terms in 2x^{2}-196+3x^{2}-21x.

$$\frac{5x^{2}-196-21x}{x\left(x-7\right)\left(x+7\right)}$$

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

$$\frac{5x^{2}-196-21x}{x^{3}-49x}$$

Differentiate w.r.t. x

$$\frac{-5x^{4}+42x^{3}+343x^{2}-9604}{\left(x\left(x^{2}-49\right)\right)^{2}}$$