Algebra Examples

$\frac{x + 3}{x^{2} + 7 x + 12} \cdot \frac{x^{2} + x - 12}{3 x - 18}$

Step 1

Factor $x^{2} + 7 x + 12$ 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 $12$ and whose sum is $7$ .

$3, 4$

Step 1.2

Write the factored form using these integers.

$\frac{x + 3}{(x + 3) (x + 4)} \cdot \frac{x^{2} + x - 12}{3 x - 18}$

Step 2

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 2.2

Write the factored form using these integers.

$\frac{x + 3}{(x + 3) (x + 4)} \cdot \frac{(x - 3) (x + 4)}{3 x - 18}$

Step 3

Simplify terms.

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Step 3.1

Factor $3$ out of $3 x - 18$ .

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Step 3.1.1

Factor $3$ out of $3 x$ .

$\frac{x + 3}{(x + 3) (x + 4)} \cdot \frac{(x - 3) (x + 4)}{3 (x) - 18}$

Step 3.1.2

Factor $3$ out of $- 18$ .

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

Step 3.1.3

Factor $3$ out of $3 x + 3 \cdot - 6$ .

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

Step 3.2

Cancel the common factor of $x + 4$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $x + 4$ out of $(x - 3) (x + 4)$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $x + 3$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

$\frac{x - 3}{3 (x - 6)}$