Solve 7/8x^2-18+1/2x^2+3x-1/4x-6

$$\frac { 7 } { 8 x ^ { 2 } - 18 } + \frac { 1 } { 2 x ^ { 2 } + 3 x } - \frac { 1 } { 4 x - 6 }$$

Evaluate

$$\frac{\left(1-x\right)\left(x-3\right)}{x\left(4x^{2}-9\right)}$$

Short Solution Steps

Factor 8x^{2}-18. Factor 2x^{2}+3x.

$$\frac{7}{2\left(2x-3\right)\left(2x+3\right)}+\frac{1}{x\left(2x+3\right)}-\frac{1}{4x-6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(2x-3\right)\left(2x+3\right) and x\left(2x+3\right) is 2x\left(2x-3\right)\left(2x+3\right). Multiply \frac{7}{2\left(2x-3\right)\left(2x+3\right)} times \frac{x}{x}. Multiply \frac{1}{x\left(2x+3\right)} times \frac{2\left(2x-3\right)}{2\left(2x-3\right)}.

$$\frac{7x}{2x\left(2x-3\right)\left(2x+3\right)}+\frac{2\left(2x-3\right)}{2x\left(2x-3\right)\left(2x+3\right)}-\frac{1}{4x-6}$$

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

$$\frac{7x+2\left(2x-3\right)}{2x\left(2x-3\right)\left(2x+3\right)}-\frac{1}{4x-6}$$

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

$$\frac{7x+4x-6}{2x\left(2x-3\right)\left(2x+3\right)}-\frac{1}{4x-6}$$

Combine like terms in 7x+4x-6.

$$\frac{11x-6}{2x\left(2x-3\right)\left(2x+3\right)}-\frac{1}{4x-6}$$

Factor 4x-6.

$$\frac{11x-6}{2x\left(2x-3\right)\left(2x+3\right)}-\frac{1}{2\left(2x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x\left(2x-3\right)\left(2x+3\right) and 2\left(2x-3\right) is 2x\left(2x-3\right)\left(2x+3\right). Multiply \frac{1}{2\left(2x-3\right)} times \frac{x\left(2x+3\right)}{x\left(2x+3\right)}.

$$\frac{11x-6}{2x\left(2x-3\right)\left(2x+3\right)}-\frac{x\left(2x+3\right)}{2x\left(2x-3\right)\left(2x+3\right)}$$

Since \frac{11x-6}{2x\left(2x-3\right)\left(2x+3\right)} and \frac{x\left(2x+3\right)}{2x\left(2x-3\right)\left(2x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{11x-6-x\left(2x+3\right)}{2x\left(2x-3\right)\left(2x+3\right)}$$

Do the multiplications in 11x-6-x\left(2x+3\right).

$$\frac{11x-6-2x^{2}-3x}{2x\left(2x-3\right)\left(2x+3\right)}$$

Combine like terms in 11x-6-2x^{2}-3x.

$$\frac{8x-6-2x^{2}}{2x\left(2x-3\right)\left(2x+3\right)}$$

Factor the expressions that are not already factored in \frac{8x-6-2x^{2}}{2x\left(2x-3\right)\left(2x+3\right)}.

$$\frac{2\left(x-3\right)\left(-x+1\right)}{2x\left(2x-3\right)\left(2x+3\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\left(x-3\right)\left(-x+1\right)}{x\left(2x-3\right)\left(2x+3\right)}$$

Expand x\left(2x-3\right)\left(2x+3\right).

$$\frac{\left(x-3\right)\left(-x+1\right)}{4x^{3}-9x}$$

Use the distributive property to multiply x-3 by -x+1 and combine like terms.

$$\frac{-x^{2}+4x-3}{4x^{3}-9x}$$

Factor

$$\frac{\left(1-x\right)\left(x-3\right)}{x\left(4x^{2}-9\right)}$$