Algebra Examples

$\frac{x^{2} - 2 x - 3}{15 - 2 x - x^{2}}$

Step 1

Factor $x^{2} - 2 x - 3$ using the AC method.

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Step 1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $15$ as $16$ plus $- 1$

$\frac{(x - 3) (x + 1)}{16 - 1 - 2 x - x^{2}}$

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{(x - 3) (x + 1)}{16 - (1^{2} + 2 x + x^{2})}$

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot 1 \cdot x$

Step 2.2.3

Rewrite the polynomial.

$\frac{(x - 3) (x + 1)}{16 - (1^{2} + 2 \cdot 1 \cdot x + x^{2})}$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 1$ and $b = x$ .

$\frac{(x - 3) (x + 1)}{16 - {(1 + x)}^{2}}$

Step 2.3

Rewrite $16$ as $4^{2}$ .

$\frac{(x - 3) (x + 1)}{4^{2} - {(1 + x)}^{2}}$

Step 2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 1 + x$ .

$\frac{(x - 3) (x + 1)}{(4 + 1 + x) (4 - (1 + x))}$

Step 2.5

Simplify.

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Step 2.5.1

Add $4$ and $1$ .

$\frac{(x - 3) (x + 1)}{(5 + x) (4 - (1 + x))}$

Step 2.5.2

Apply the distributive property.

$\frac{(x - 3) (x + 1)}{(5 + x) (4 - 1 \cdot 1 - x)}$

Step 2.5.3

Multiply $- 1$ by $1$ .

$\frac{(x - 3) (x + 1)}{(5 + x) (4 - 1 - x)}$

Step 2.5.4

Subtract $1$ from $4$ .

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $x - 3$ and $3 - x$ .

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Step 3.1.1

Factor $- 1$ out of $x$ .

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

Step 3.1.2

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 3.1.3

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

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

Step 3.1.4

Reorder terms.

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

Step 3.1.5

Cancel the common factor.

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

Step 3.1.6

Rewrite the expression.

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

Step 3.2

Move the negative in front of the fraction.

$- \frac{x + 1}{5 + x}$