Pre-Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

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 $- 2$ .

$- 3, 1$

Step 2.2

Write the factored form using these integers.

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

Step 3

Factor $4$ out of $4 x - 4$ .

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Step 3.1

Factor $4$ out of $4 x$ .

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

Step 3.2

Factor $4$ out of $- 4$ .

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

Step 3.3

Factor $4$ out of $4 (x) + 4 (- 1)$ .

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

Step 4

Cancel the common factor of $x - 1$ .

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Step 4.1

Factor $x - 1$ out of $4 (x - 1)$ .

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Factor $x^{2} + 4 x + 3$ using the AC method.

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Step 6.1

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 $4$ .

$1, 3$

Step 6.2

Write the factored form using these integers.

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

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

Step 7

Cancel the common factor of $x + 1$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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