Simplify (3x-1)/(x^2+2x-15)-2/(x+5)

$\frac{3 x - 1}{x^{2} + 2 x - 15} - \frac{2}{x + 5}$

Step 1

Factor $x^{2} + 2 x - 15$ 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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Write the factored form using these integers.

$\frac{3 x - 1}{(x - 3) (x + 5)} - \frac{2}{x + 5}$

Step 2

To write $- \frac{2}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{3 x - 1}{(x - 3) (x + 5)} - \frac{2}{x + 5} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of $(x - 3) (x + 5)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{x + 5}$ by $\frac{x - 3}{x - 3}$ .

$\frac{3 x - 1}{(x - 3) (x + 5)} - \frac{2 (x - 3)}{(x + 5) (x - 3)}$

Reorder the factors of $(x - 3) (x + 5)$ .

$\frac{3 x - 1}{(x + 5) (x - 3)} - \frac{2 (x - 3)}{(x + 5) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 x - 1 - 2 (x - 3)}{(x + 5) (x - 3)}$

Step 5

Simplify the numerator.

Apply the distributive property.

$\frac{3 x - 1 - 2 x - 2 \cdot - 3}{(x + 5) (x - 3)}$

Multiply $- 2$ by $- 3$ .

$\frac{3 x - 1 - 2 x + 6}{(x + 5) (x - 3)}$

Subtract $2 x$ from $3 x$ .

$\frac{x - 1 + 6}{(x + 5) (x - 3)}$

Add $- 1$ and $6$ .

$\frac{x + 5}{(x + 5) (x - 3)}$

Step 6

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{x + 5}{(x + 5) (x - 3)}$

Rewrite the expression.

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

Simplify (3x-1)/(x^2+2x-15)-2/(x+5)

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