Simplify (3x^2-1)/(x^2-3x-18)-(x+2)/(x-6)

$\frac{3 x^{2} - 1}{x^{2} - 3 x - 18} - \frac{x + 2}{x - 6}$

Step 1

Factor $x^{2} - 3 x - 18$ 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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Write the factored form using these integers.

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

Step 2

To write $- \frac{x + 2}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 3

Simplify terms.

Multiply $\frac{x + 2}{x - 6}$ by $\frac{x + 3}{x + 3}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Expand $(- x - 2) (x + 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $- 1$ .

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

Multiply $- 2$ by $3$ .

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

Subtract $2 x$ from $- 3 x$ .

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

Subtract $x^{2}$ from $3 x^{2}$ .

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

Subtract $6$ from $- 1$ .

$\frac{2 x^{2} - 5 x - 7}{(x - 6) (x + 3)}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 7 = - 14$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$\frac{2 x^{2} - 5 (x) - 7}{(x - 6) (x + 3)}$

Rewrite $- 5$ as $2$ plus $- 7$

$\frac{2 x^{2} + (2 - 7) x - 7}{(x - 6) (x + 3)}$

Apply the distributive property.

$\frac{2 x^{2} + 2 x - 7 x - 7}{(x - 6) (x + 3)}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 x^{2} + 2 x) - 7 x - 7}{(x - 6) (x + 3)}$

Factor out the greatest common factor (GCF) from each group.

$\frac{2 x (x + 1) - 7 (x + 1)}{(x - 6) (x + 3)}$

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$\frac{(x + 1) (2 x - 7)}{(x - 6) (x + 3)}$

Simplify (3x^2-1)/(x^2-3x-18)-(x+2)/(x-6)

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