Simplify (5x+1)/(x-6)-x/(x^2-10x+24)

$\frac{5 x + 1}{x - 6} - \frac{x}{x^{2} - 10 x + 24}$

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.

$\frac{5 x + 1}{x - 6} - \frac{x}{(x - 6) (x - 4)}$

Step 2

To write $\frac{5 x + 1}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{5 x + 1}{x - 6} \cdot \frac{x - 4}{x - 4} - \frac{x}{(x - 6) (x - 4)}$

Step 3

Simplify terms.

Multiply $\frac{5 x + 1}{x - 6}$ by $\frac{x - 4}{x - 4}$ .

$\frac{(5 x + 1) (x - 4)}{(x - 6) (x - 4)} - \frac{x}{(x - 6) (x - 4)}$

Combine the numerators over the common denominator.

$\frac{(5 x + 1) (x - 4) - x}{(x - 6) (x - 4)}$

Step 4

Simplify the numerator.

Expand $(5 x + 1) (x - 4)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 x (x - 4) + 1 (x - 4) - x}{(x - 6) (x - 4)}$

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot - 4 + 1 (x - 4) - x}{(x - 6) (x - 4)}$

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot - 4 + 1 x + 1 \cdot - 4 - x}{(x - 6) (x - 4)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{5 (x \cdot x) + 5 x \cdot - 4 + 1 x + 1 \cdot - 4 - x}{(x - 6) (x - 4)}$

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 5 x \cdot - 4 + 1 x + 1 \cdot - 4 - x}{(x - 6) (x - 4)}$

Multiply $- 4$ by $5$ .

$\frac{5 x^{2} - 20 x + 1 x + 1 \cdot - 4 - x}{(x - 6) (x - 4)}$

Multiply $x$ by $1$ .

$\frac{5 x^{2} - 20 x + x + 1 \cdot - 4 - x}{(x - 6) (x - 4)}$

Multiply $- 4$ by $1$ .

$\frac{5 x^{2} - 20 x + x - 4 - x}{(x - 6) (x - 4)}$

Add $- 20 x$ and $x$ .

$\frac{5 x^{2} - 19 x - 4 - x}{(x - 6) (x - 4)}$

Subtract $x$ from $- 19 x$ .

$\frac{5 x^{2} - 20 x - 4}{(x - 6) (x - 4)}$

Simplify (5x+1)/(x-6)-x/(x^2-10x+24)

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