Precalculus Examples

$\frac{\frac{x^{2} + 2 x - 3}{x^{2} + 8 x + 16}}{\frac{x - 1}{3 x + 12}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 2 x - 3}{x^{2} + 8 x + 16} \cdot \frac{3 x + 12}{x - 1}$

Step 2

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

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Step 2.1

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 $2$ .

$- 1, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 1) (x + 3)}{x^{2} + 8 x + 16} \cdot \frac{3 x + 12}{x - 1}$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$\frac{(x - 1) (x + 3)}{x^{2} + 8 x + 4^{2}} \cdot \frac{3 x + 12}{x - 1}$

Step 3.2

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

$8 x = 2 \cdot x \cdot 4$

Step 3.3

Rewrite the polynomial.

$\frac{(x - 1) (x + 3)}{x^{2} + 2 \cdot x \cdot 4 + 4^{2}} \cdot \frac{3 x + 12}{x - 1}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$\frac{(x - 1) (x + 3)}{{(x + 4)}^{2}} \cdot \frac{3 x + 12}{x - 1}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 1$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(x - 1) (x + 3)}{{(x + 4)}^{2}} \cdot \frac{3 x + 12}{x - 1}$

Step 4.1.2

Rewrite the expression.

$\frac{x + 3}{{(x + 4)}^{2}} (3 x + 12)$

Step 4.2

Multiply $\frac{x + 3}{{(x + 4)}^{2}}$ by $3 x + 12$ .

$\frac{(x + 3) (3 x + 12)}{{(x + 4)}^{2}}$

Step 5

Factor $3$ out of $3 x + 12$ .

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Step 5.1

Factor $3$ out of $3 x$ .

$\frac{(x + 3) (3 (x) + 12)}{{(x + 4)}^{2}}$

Step 5.2

Factor $3$ out of $12$ .

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

Step 5.3

Factor $3$ out of $3 x + 3 \cdot 4$ .

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

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

Step 6

Cancel the common factor of $x + 4$ and ${(x + 4)}^{2}$ .

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Step 6.1

Factor $x + 4$ out of $(x + 3) \cdot 3 (x + 4)$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $x + 4$ out of ${(x + 4)}^{2}$ .

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

Move $3$ to the left of $x + 3$ .

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