Algebra Examples

$\frac{6 - x - x^{2}}{x^{2} - 9}$

Step 1

Factor by grouping.

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Step 1.1

Reorder terms.

$\frac{- x^{2} - x + 6}{x^{2} - 9}$

Step 1.2

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

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Step 1.2.1

Factor $- 1$ out of $- x$ .

$\frac{- x^{2} - (x) + 6}{x^{2} - 9}$

Step 1.2.2

Rewrite $- 1$ as $2$ plus $- 3$

$\frac{- x^{2} + (2 - 3) x + 6}{x^{2} - 9}$

Step 1.2.3

Apply the distributive property.

$\frac{- x^{2} + 2 x - 3 x + 6}{x^{2} - 9}$

Step 1.3

Factor out the greatest common factor from each group.

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Step 1.3.1

Group the first two terms and the last two terms.

$\frac{(- x^{2} + 2 x) - 3 x + 6}{x^{2} - 9}$

Step 1.3.2

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

$\frac{x (- x + 2) + 3 (- x + 2)}{x^{2} - 9}$

Step 1.4

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

$\frac{(- x + 2) (x + 3)}{x^{2} - 9}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Factor $- 1$ out of $- x$ .

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

Step 3.3

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 3.4

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

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

Step 3.5

Simplify the expression.

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Step 3.5.1

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 3.5.2

Move the negative in front of the fraction.

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