Algebra Examples

$\frac{\frac{4 x^{2} + 12 x - 16}{2 x + 10}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1

Cancel the common factor of $4 x^{2} + 12 x - 16$ and $2 x + 10$ .

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Step 1.1

Factor $2$ out of $4 x^{2}$ .

$\frac{\frac{2 (2 x^{2}) + 12 x - 16}{2 x + 10}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.2

Factor $2$ out of $12 x$ .

$\frac{\frac{2 (2 x^{2}) + 2 (6 x) - 16}{2 x + 10}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.3

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

$\frac{\frac{2 (2 x^{2} + 6 x) - 16}{2 x + 10}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.4

Factor $2$ out of $- 16$ .

$\frac{\frac{2 (2 x^{2} + 6 x) + 2 \cdot - 8}{2 x + 10}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.5

Factor $2$ out of $2 (2 x^{2} + 6 x) + 2 (- 8)$ .

$\frac{\frac{2 (2 x^{2} + 6 x - 8)}{2 x + 10}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $2$ out of $2 x$ .

$\frac{\frac{2 (2 x^{2} + 6 x - 8)}{2 (x) + 10}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.6.2

Factor $2$ out of $10$ .

$\frac{\frac{2 (2 x^{2} + 6 x - 8)}{2 x + 2 \cdot 5}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.6.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{\frac{2 (2 x^{2} + 6 x - 8)}{2 (x + 5)}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.6.4

Cancel the common factor.

$\frac{\frac{2 (2 x^{2} + 6 x - 8)}{2 (x + 5)}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 1.6.5

Rewrite the expression.

$\frac{\frac{2 x^{2} + 6 x - 8}{x + 5}}{\frac{6 x + 24}{x^{2} + 9 x + 20}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 x^{2} + 6 x - 8}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $2$ out of $2 x^{2} + 6 x - 8$ .

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Step 3.1.1

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

$\frac{2 (x^{2}) + 6 x - 8}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

Step 3.1.2

Factor $2$ out of $6 x$ .

$\frac{2 (x^{2}) + 2 (3 x) - 8}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

Step 3.1.3

Factor $2$ out of $- 8$ .

$\frac{2 x^{2} + 2 (3 x) + 2 \cdot - 4}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

Step 3.1.4

Factor $2$ out of $2 x^{2} + 2 (3 x)$ .

$\frac{2 (x^{2} + 3 x) + 2 \cdot - 4}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

Step 3.1.5

Factor $2$ out of $2 (x^{2} + 3 x) + 2 \cdot - 4$ .

$\frac{2 (x^{2} + 3 x - 4)}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

Step 3.2

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

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Step 3.2.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 3.2.2

Write the factored form using these integers.

$\frac{2 ((x - 1) (x + 4))}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

$\frac{2 (x - 1) (x + 4)}{x + 5} \cdot \frac{x^{2} + 9 x + 20}{6 x + 24}$

Step 4

Factor $x^{2} + 9 x + 20$ 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 $20$ and whose sum is $9$ .

$4, 5$

Step 4.2

Write the factored form using these integers.

$\frac{2 (x - 1) (x + 4)}{x + 5} \cdot \frac{(x + 4) (x + 5)}{6 x + 24}$

Step 5

Simplify terms.

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Step 5.1

Factor $6$ out of $6 x + 24$ .

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Step 5.1.1

Factor $6$ out of $6 x$ .

$\frac{2 (x - 1) (x + 4)}{x + 5} \cdot \frac{(x + 4) (x + 5)}{6 (x) + 24}$

Step 5.1.2

Factor $6$ out of $24$ .

$\frac{2 (x - 1) (x + 4)}{x + 5} \cdot \frac{(x + 4) (x + 5)}{6 x + 6 \cdot 4}$

Step 5.1.3

Factor $6$ out of $6 x + 6 \cdot 4$ .

$\frac{2 (x - 1) (x + 4)}{x + 5} \cdot \frac{(x + 4) (x + 5)}{6 (x + 4)}$

Step 5.2

Simplify terms.

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Step 5.2.1

Cancel the common factor of $2 (x + 4)$ .

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Step 5.2.1.1

Factor $2 (x + 4)$ out of $2 (x - 1) (x + 4)$ .

$\frac{2 (x + 4) (x - 1)}{x + 5} \cdot \frac{(x + 4) (x + 5)}{6 (x + 4)}$

Step 5.2.1.2

Factor $2 (x + 4)$ out of $6 (x + 4)$ .

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

Step 5.2.1.3

Cancel the common factor.

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

Step 5.2.1.4

Rewrite the expression.

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

Step 5.2.2

Cancel the common factor of $x + 5$ .

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Step 5.2.2.1

Factor $x + 5$ out of $(x + 4) (x + 5)$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.3

Apply the distributive property.

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

Step 5.2.4

Combine $x$ and $\frac{x + 4}{3}$ .

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

Step 5.2.5

Rewrite $- 1 \frac{x + 4}{3}$ as $- \frac{x + 4}{3}$ .

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

Step 5.2.6

Combine the numerators over the common denominator.

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

Step 5.3

Simplify each term.

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Step 5.3.1

Apply the distributive property.

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

Step 5.3.2

Multiply $x$ by $x$ .

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

Step 5.3.3

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

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

Step 5.3.4

Apply the distributive property.

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

Step 5.3.5

Multiply $- 1$ by $4$ .

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

Step 5.4

Subtract $x$ from $4 x$ .

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

Step 6

Factor $x^{2} + 3 x - 4$ 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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 6.2

Write the factored form using these integers.

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