Solve x/4x^2-12x+9-3x/9-4x^2

$$\frac { x } { 4 x ^ { 2 } - 12 x + 9 } - \frac { 3 x } { 9 - 4 x ^ { 2 } }$$

Evaluate

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

Short Solution Steps

Factor 4x^{2}-12x+9. Factor 9-4x^{2}.

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

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

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

Since \frac{x\left(2x+3\right)}{\left(2x+3\right)\left(2x-3\right)^{2}} and \frac{3x\left(-1\right)\left(2x-3\right)}{\left(2x+3\right)\left(2x-3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in x\left(2x+3\right)-3x\left(-1\right)\left(2x-3\right).

$$\frac{2x^{2}+3x+6x^{2}-9x}{\left(2x+3\right)\left(2x-3\right)^{2}}$$

Combine like terms in 2x^{2}+3x+6x^{2}-9x.

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

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

$$\frac{8x^{2}-6x}{8x^{3}-12x^{2}-18x+27}$$

Factor

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