Algebra Examples

$\frac{x + 1}{x^{2} + x - 6} \div \frac{x^{2} + 5 x + 4}{x + 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 1}{x^{2} + x - 6} \cdot \frac{x + 3}{x^{2} + 5 x + 4}$

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 $1$ .

$- 2, 3$

Step 2.2

Write the factored form using these integers.

$\frac{x + 1}{(x - 2) (x + 3)} \cdot \frac{x + 3}{x^{2} + 5 x + 4}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x + 3$ .

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Step 3.1.1

Factor $x + 3$ out of $(x - 2) (x + 3)$ .

$\frac{x + 1}{(x + 3) (x - 2)} \cdot \frac{x + 3}{x^{2} + 5 x + 4}$

Step 3.1.2

Cancel the common factor.

$\frac{x + 1}{(x + 3) (x - 2)} \cdot \frac{x + 3}{x^{2} + 5 x + 4}$

Step 3.1.3

Rewrite the expression.

$\frac{x + 1}{x - 2} \cdot \frac{1}{x^{2} + 5 x + 4}$

Step 3.2

Multiply $\frac{x + 1}{x - 2}$ by $\frac{1}{x^{2} + 5 x + 4}$ .

$\frac{x + 1}{(x - 2) (x^{2} + 5 x + 4)}$

Step 4

Factor $x^{2} + 5 x + 4$ using the AC method.

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Step 4.1

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 $4$ and whose sum is $5$ .

$1, 4$

Step 4.2

Write the factored form using these integers.

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

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

Step 5

Cancel the common factor of $x + 1$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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