Algebra Examples

$\frac{12 x + 48}{6 x - 15} \cdot \frac{4 x^{2} - 25}{x^{2} + 9 x + 20}$

Step 1

Simplify with factoring out.

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Step 1.1

Factor $12$ out of $12 x + 48$ .

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Step 1.1.1

Factor $12$ out of $12 x$ .

$\frac{12 (x) + 48}{6 x - 15} \cdot \frac{4 x^{2} - 25}{x^{2} + 9 x + 20}$

Step 1.1.2

Factor $12$ out of $48$ .

$\frac{12 x + 12 \cdot 4}{6 x - 15} \cdot \frac{4 x^{2} - 25}{x^{2} + 9 x + 20}$

Step 1.1.3

Factor $12$ out of $12 x + 12 \cdot 4$ .

$\frac{12 (x + 4)}{6 x - 15} \cdot \frac{4 x^{2} - 25}{x^{2} + 9 x + 20}$

Step 1.2

Factor $3$ out of $6 x - 15$ .

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Step 1.2.1

Factor $3$ out of $6 x$ .

$\frac{12 (x + 4)}{3 (2 x) - 15} \cdot \frac{4 x^{2} - 25}{x^{2} + 9 x + 20}$

Step 1.2.2

Factor $3$ out of $- 15$ .

$\frac{12 (x + 4)}{3 (2 x) + 3 (- 5)} \cdot \frac{4 x^{2} - 25}{x^{2} + 9 x + 20}$

Step 1.2.3

Factor $3$ out of $3 (2 x) + 3 (- 5)$ .

$\frac{12 (x + 4)}{3 (2 x - 5)} \cdot \frac{4 x^{2} - 25}{x^{2} + 9 x + 20}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{12 (x + 4)}{3 (2 x - 5)} \cdot \frac{{(2 x)}^{2} - 25}{x^{2} + 9 x + 20}$

Step 2.2

Rewrite $25$ as $5^{2}$ .

$\frac{12 (x + 4)}{3 (2 x - 5)} \cdot \frac{{(2 x)}^{2} - 5^{2}}{x^{2} + 9 x + 20}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 5$ .

$\frac{12 (x + 4)}{3 (2 x - 5)} \cdot \frac{(2 x + 5) (2 x - 5)}{x^{2} + 9 x + 20}$

Step 3

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

$4, 5$

Step 3.2

Write the factored form using these integers.

$\frac{12 (x + 4)}{3 (2 x - 5)} \cdot \frac{(2 x + 5) (2 x - 5)}{(x + 4) (x + 5)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 4$ .

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Step 4.1.1

Factor $x + 4$ out of $12 (x + 4)$ .

$\frac{(x + 4) \cdot 12}{3 (2 x - 5)} \cdot \frac{(2 x + 5) (2 x - 5)}{(x + 4) (x + 5)}$

Step 4.1.2

Cancel the common factor.

$\frac{(x + 4) \cdot 12}{3 (2 x - 5)} \cdot \frac{(2 x + 5) (2 x - 5)}{(x + 4) (x + 5)}$

Step 4.1.3

Rewrite the expression.

$\frac{12}{3 (2 x - 5)} \cdot \frac{(2 x + 5) (2 x - 5)}{x + 5}$

Step 4.2

Cancel the common factor of $2 x - 5$ .

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Step 4.2.1

Factor $2 x - 5$ out of $3 (2 x - 5)$ .

$\frac{12}{(2 x - 5) \cdot 3} \cdot \frac{(2 x + 5) (2 x - 5)}{x + 5}$

Step 4.2.2

Factor $2 x - 5$ out of $(2 x + 5) (2 x - 5)$ .

$\frac{12}{(2 x - 5) \cdot 3} \cdot \frac{(2 x - 5) (2 x + 5)}{x + 5}$

Step 4.2.3

Cancel the common factor.

$\frac{12}{(2 x - 5) \cdot 3} \cdot \frac{(2 x - 5) (2 x + 5)}{x + 5}$

Step 4.2.4

Rewrite the expression.

$\frac{12}{3} \cdot \frac{2 x + 5}{x + 5}$

Step 4.3

Multiply $\frac{12}{3}$ by $\frac{2 x + 5}{x + 5}$ .

$\frac{12 (2 x + 5)}{3 (x + 5)}$

Step 4.4

Cancel the common factor of $12$ and $3$ .

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Step 4.4.1

Factor $3$ out of $12 (2 x + 5)$ .

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

Cancel the common factor.

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

Step 4.4.2.2

Rewrite the expression.

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