Solve x-1/x^2-4-3^2/x^2-4x+4

$$\frac { x - 1 } { x ^ { 2 } - 4 } - \frac { 3 ^ { 2 } } { x ^ { 2 } - 4 x + 4 }$$

Evaluate

$$\frac{x^{2}-12x-16}{\left(x+2\right)\left(x-2\right)^{2}}$$

Short Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\frac{x-1}{x^{2}-4}-\frac{9}{x^{2}-4x+4}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and \left(x-2\right)^{2} is \left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{x-1}{\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{9}{\left(x-2\right)^{2}} times \frac{x+2}{x+2}.

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

Since \frac{\left(x-1\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}} and \frac{9\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-1\right)\left(x-2\right)-9\left(x+2\right).

$$\frac{x^{2}-2x-x+2-9x-18}{\left(x+2\right)\left(x-2\right)^{2}}$$

Combine like terms in x^{2}-2x-x+2-9x-18.

$$\frac{x^{2}-12x-16}{\left(x+2\right)\left(x-2\right)^{2}}$$

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

$$\frac{x^{2}-12x-16}{x^{3}-2x^{2}-4x+8}$$

Expand

$$\frac{x^{2}-12x-16}{\left(x+2\right)\left(x-2\right)^{2}}$$

Short Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\frac{x-1}{x^{2}-4}-\frac{9}{x^{2}-4x+4}$$

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and \left(x-2\right)^{2} is \left(x+2\right)\left(x-2\right)^{2}. Multiply \frac{x-1}{\left(x-2\right)\left(x+2\right)} times \frac{x-2}{x-2}. Multiply \frac{9}{\left(x-2\right)^{2}} times \frac{x+2}{x+2}.

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

Since \frac{\left(x-1\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)^{2}} and \frac{9\left(x+2\right)}{\left(x+2\right)\left(x-2\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x-1\right)\left(x-2\right)-9\left(x+2\right).

$$\frac{x^{2}-2x-x+2-9x-18}{\left(x+2\right)\left(x-2\right)^{2}}$$

Combine like terms in x^{2}-2x-x+2-9x-18.

$$\frac{x^{2}-12x-16}{\left(x+2\right)\left(x-2\right)^{2}}$$

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

$$\frac{x^{2}-12x-16}{x^{3}-2x^{2}-4x+8}$$