Simplify 1/(x^2-10x+24)-(2x)/(x-6)

$\frac{1}{x^{2} - 10 x + 24} - \frac{2 x}{x - 6}$

Step 1

Factor $x^{2} - 10 x + 24$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $24$ and whose sum is $- 10$ .

$- 6, - 4$

Write the factored form using these integers.

$\frac{1}{(x - 6) (x - 4)} - \frac{2 x}{x - 6}$

Step 2

To write $- \frac{2 x}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{1}{(x - 6) (x - 4)} - \frac{2 x}{x - 6} \cdot \frac{x - 4}{x - 4}$

Step 3

Simplify terms.

Multiply $\frac{2 x}{x - 6}$ by $\frac{x - 4}{x - 4}$ .

$\frac{1}{(x - 6) (x - 4)} - \frac{2 x (x - 4)}{(x - 6) (x - 4)}$

Combine the numerators over the common denominator.

$\frac{1 - 2 x (x - 4)}{(x - 6) (x - 4)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{1 - 2 x \cdot x - 2 x \cdot - 4}{(x - 6) (x - 4)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{1 - 2 (x \cdot x) - 2 x \cdot - 4}{(x - 6) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{1 - 2 x^{2} - 2 x \cdot - 4}{(x - 6) (x - 4)}$

Multiply $- 4$ by $- 2$ .

$\frac{1 - 2 x^{2} + 8 x}{(x - 6) (x - 4)}$

Reorder terms.

$\frac{- 2 x^{2} + 8 x + 1}{(x - 6) (x - 4)}$

Step 5

Simplify with factoring out.

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{- (2 x^{2}) + 8 x + 1}{(x - 6) (x - 4)}$

Factor $- 1$ out of $8 x$ .

$\frac{- (2 x^{2}) - (- 8 x) + 1}{(x - 6) (x - 4)}$

Factor $- 1$ out of $- (2 x^{2}) - (- 8 x)$ .

$\frac{- (2 x^{2} - 8 x) + 1}{(x - 6) (x - 4)}$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (2 x^{2} - 8 x) - 1 (- 1)}{(x - 6) (x - 4)}$

Factor $- 1$ out of $- (2 x^{2} - 8 x) - 1 (- 1)$ .

$\frac{- (2 x^{2} - 8 x - 1)}{(x - 6) (x - 4)}$

Simplify the expression.

Rewrite $- (2 x^{2} - 8 x - 1)$ as $- 1 (2 x^{2} - 8 x - 1)$ .

$\frac{- 1 (2 x^{2} - 8 x - 1)}{(x - 6) (x - 4)}$

Move the negative in front of the fraction.

$- \frac{2 x^{2} - 8 x - 1}{(x - 6) (x - 4)}$

Simplify 1/(x^2-10x+24)-(2x)/(x-6)

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