Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

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.

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

Step 3

Cancel the common factor of $x + 1$ .

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Step 3.1

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

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

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

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Step 5.1

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 5.2

Write the factored form using these integers.

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

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

Step 6

Cancel the common factor of $x + 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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