Solve 8/x+6+12-x/x-6-1

$$\frac{ 8 }{ x+6 } + \frac{ 12-x }{ x-6 } -1$$

Evaluate

$$-\frac{2\left(x-10\right)\left(x+3\right)}{x^{2}-36}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+6 and x-6 is \left(x-6\right)\left(x+6\right). Multiply \frac{8}{x+6} times \frac{x-6}{x-6}. Multiply \frac{12-x}{x-6} times \frac{x+6}{x+6}.

$$\frac{8\left(x-6\right)}{\left(x-6\right)\left(x+6\right)}+\frac{\left(12-x\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}-1$$

Since \frac{8\left(x-6\right)}{\left(x-6\right)\left(x+6\right)} and \frac{\left(12-x\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{8\left(x-6\right)+\left(12-x\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}-1$$

Do the multiplications in 8\left(x-6\right)+\left(12-x\right)\left(x+6\right).

$$\frac{8x-48+12x+72-x^{2}-6x}{\left(x-6\right)\left(x+6\right)}-1$$

Combine like terms in 8x-48+12x+72-x^{2}-6x.

$$\frac{14x+24-x^{2}}{\left(x-6\right)\left(x+6\right)}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

$$\frac{14x+24-x^{2}}{\left(x-6\right)\left(x+6\right)}-\frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}$$

Since \frac{14x+24-x^{2}}{\left(x-6\right)\left(x+6\right)} and \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{14x+24-x^{2}-\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}$$

Do the multiplications in 14x+24-x^{2}-\left(x-6\right)\left(x+6\right).

$$\frac{14x+24-x^{2}-x^{2}-6x+6x+36}{\left(x-6\right)\left(x+6\right)}$$

Combine like terms in 14x+24-x^{2}-x^{2}-6x+6x+36.

$$\frac{14x+60-2x^{2}}{\left(x-6\right)\left(x+6\right)}$$

Expand \left(x-6\right)\left(x+6\right).

$$\frac{14x+60-2x^{2}}{x^{2}-36}$$

Expand

$$-\frac{2\left(x^{2}-7x-30\right)}{x^{2}-36}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+6 and x-6 is \left(x-6\right)\left(x+6\right). Multiply \frac{8}{x+6} times \frac{x-6}{x-6}. Multiply \frac{12-x}{x-6} times \frac{x+6}{x+6}.

$$\frac{8\left(x-6\right)}{\left(x-6\right)\left(x+6\right)}+\frac{\left(12-x\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}-1$$

Since \frac{8\left(x-6\right)}{\left(x-6\right)\left(x+6\right)} and \frac{\left(12-x\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{8\left(x-6\right)+\left(12-x\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}-1$$

Do the multiplications in 8\left(x-6\right)+\left(12-x\right)\left(x+6\right).

$$\frac{8x-48+12x+72-x^{2}-6x}{\left(x-6\right)\left(x+6\right)}-1$$

Combine like terms in 8x-48+12x+72-x^{2}-6x.

$$\frac{14x+24-x^{2}}{\left(x-6\right)\left(x+6\right)}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

$$\frac{14x+24-x^{2}}{\left(x-6\right)\left(x+6\right)}-\frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}$$

Since \frac{14x+24-x^{2}}{\left(x-6\right)\left(x+6\right)} and \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{14x+24-x^{2}-\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}$$

Do the multiplications in 14x+24-x^{2}-\left(x-6\right)\left(x+6\right).

$$\frac{14x+24-x^{2}-x^{2}-6x+6x+36}{\left(x-6\right)\left(x+6\right)}$$

Combine like terms in 14x+24-x^{2}-x^{2}-6x+6x+36.

$$\frac{14x+60-2x^{2}}{\left(x-6\right)\left(x+6\right)}$$

Expand \left(x-6\right)\left(x+6\right).

$$\frac{14x+60-2x^{2}}{x^{2}-36}$$