Algebra Examples

$\frac{10}{4 x - 8} \div \frac{2 x^{2} + 6 x}{x^{2} + x - 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{10}{4 x - 8} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 2

Simplify terms.

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Step 2.1

Cancel the common factor of $10$ and $4 x - 8$ .

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Step 2.1.1

Factor $2$ out of $10$ .

$\frac{2 (5)}{4 x - 8} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $2$ out of $4 x$ .

$\frac{2 (5)}{2 (2 x) - 8} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 2.1.2.2

Factor $2$ out of $- 8$ .

$\frac{2 \cdot 5}{2 (2 x) + 2 \cdot - 4} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 2.1.2.3

Factor $2$ out of $2 (2 x) + 2 (- 4)$ .

$\frac{2 (5)}{2 (2 x - 4)} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 2.1.2.4

Cancel the common factor.

$\frac{2 \cdot 5}{2 (2 x - 4)} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 2.1.2.5

Rewrite the expression.

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

Step 2.2

Factor $2$ out of $2 x - 4$ .

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Step 2.2.1

Factor $2$ out of $2 x$ .

$\frac{5}{2 (x) - 4} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 2.2.2

Factor $2$ out of $- 4$ .

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

Step 2.2.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$\frac{5}{2 (x - 2)} \cdot \frac{x^{2} + x - 6}{2 x^{2} + 6 x}$

Step 3

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

$- 2, 3$

Step 3.2

Write the factored form using these integers.

$\frac{5}{2 (x - 2)} \cdot \frac{(x - 2) (x + 3)}{2 x^{2} + 6 x}$

Step 4

Simplify terms.

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Step 4.1

Factor $2 x$ out of $2 x^{2} + 6 x$ .

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Step 4.1.1

Factor $2 x$ out of $2 x^{2}$ .

$\frac{5}{2 (x - 2)} \cdot \frac{(x - 2) (x + 3)}{2 x (x) + 6 x}$

Step 4.1.2

Factor $2 x$ out of $6 x$ .

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

Step 4.1.3

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

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

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

$\frac{5}{2} \cdot \frac{x + 3}{2 x (x + 3)}$

Step 4.3

Multiply $\frac{5}{2}$ by $\frac{x + 3}{2 x (x + 3)}$ .

$\frac{5 (x + 3)}{2 (2 x (x + 3))}$

Step 4.4

Multiply $2$ by $2$ .

$\frac{5 (x + 3)}{4 (x (x + 3))}$

Step 4.5

Cancel the common factor of $x + 3$ .

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Step 4.5.1

Cancel the common factor.

$\frac{5 (x + 3)}{4 (x (x + 3))}$

Step 4.5.2

Rewrite the expression.

$\frac{5}{4 (x)}$

$\frac{5}{4 x}$