Pre-Algebra Examples

$\frac{x^{2} - x - 6}{x^{2} + 2 x - 3} \div \frac{x - 3}{4 x + 12}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - x - 6}{x^{2} + 2 x - 3} \cdot \frac{4 x + 12}{x - 3}$

Step 2

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

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

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Step 3.1

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 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 3$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Multiply $\frac{x + 2}{(x - 1) (x + 3)}$ by $4 x + 12$ .

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

Step 5

Factor $4$ out of $4 x + 12$ .

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Step 5.1

Factor $4$ out of $4 x$ .

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

Step 5.2

Factor $4$ out of $12$ .

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

Step 5.3

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

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

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

Step 6

Cancel the common factor of $x + 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Move $4$ to the left of $x + 2$ .

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