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

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

Step 1

Factor $x^{2} + 5 x - 6$ 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 $- 6$ and whose sum is $5$ .

$- 1, 6$

Write the factored form using these integers.

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(x - 1) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

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

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

Reorder the factors of $(x - 1) (x + 6)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $- 1$ .

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

Subtract $x$ from $5 x$ .

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

Add $1$ and $1$ .

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

Factor $2$ out of $4 x + 2$ .

Factor $2$ out of $4 x$ .

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

Factor $2$ out of $2$ .

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

Factor $2$ out of $2 (2 x) + 2 (1)$ .

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

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

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