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

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

Step 1

Factor $x^{2} - x - 6$ 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 $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

Step 2

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 $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = 1$ .

Multiply by $1$ .

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

Rewrite $1$ as $- 3$ plus $4$

$\frac{(x - 3) (x + 2)}{2 x^{2} + (- 3 + 4) x - 6}$

Apply the distributive property.

$\frac{(x - 3) (x + 2)}{2 x^{2} - 3 x + 4 x - 6}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(x - 3) (x + 2)}{(2 x^{2} - 3 x) + 4 x - 6}$

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

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

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

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

Step 3

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x - 3}{2 x - 3}$

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

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