Simplify (x^2+5x-34)/(x^2-9x+8)-10/(x-8)

$\frac{x^{2} + 5 x - 34}{x^{2} - 9 x + 8} - \frac{10}{x - 8}$

Step 1

Factor $x^{2} - 9 x + 8$ 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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Write the factored form using these integers.

$\frac{x^{2} + 5 x - 34}{(x - 8) (x - 1)} - \frac{10}{x - 8}$

Step 2

To write $- \frac{10}{x - 8}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{x^{2} + 5 x - 34}{(x - 8) (x - 1)} - \frac{10}{x - 8} \cdot \frac{x - 1}{x - 1}$

Step 3

Simplify terms.

Multiply $\frac{10}{x - 8}$ by $\frac{x - 1}{x - 1}$ .

$\frac{x^{2} + 5 x - 34}{(x - 8) (x - 1)} - \frac{10 (x - 1)}{(x - 8) (x - 1)}$

Combine the numerators over the common denominator.

$\frac{x^{2} + 5 x - 34 - 10 (x - 1)}{(x - 8) (x - 1)}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{x^{2} + 5 x - 34 - 10 x - 10 \cdot - 1}{(x - 8) (x - 1)}$

Multiply $- 10$ by $- 1$ .

$\frac{x^{2} + 5 x - 34 - 10 x + 10}{(x - 8) (x - 1)}$

Subtract $10 x$ from $5 x$ .

$\frac{x^{2} - 5 x - 34 + 10}{(x - 8) (x - 1)}$

Add $- 34$ and $10$ .

$\frac{x^{2} - 5 x - 24}{(x - 8) (x - 1)}$

Factor $x^{2} - 5 x - 24$ 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 $- 5$ .

$- 8, 3$

Write the factored form using these integers.

$\frac{(x - 8) (x + 3)}{(x - 8) (x - 1)}$

Step 5

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{(x - 8) (x + 3)}{(x - 8) (x - 1)}$

Rewrite the expression.

$\frac{x + 3}{x - 1}$

Simplify (x^2+5x-34)/(x^2-9x+8)-10/(x-8)

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