Algebra Examples

$\frac{x^{2} + 8 x + 15}{x - 4} \cdot \frac{x^{2} - 16}{2 x + 6}$

Step 1

Factor $x^{2} + 8 x + 15$ using the AC method.

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Step 1.1

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 $15$ and whose sum is $8$ .

$3, 5$

Step 1.2

Write the factored form using these integers.

$\frac{(x + 3) (x + 5)}{x - 4} \cdot \frac{x^{2} - 16}{2 x + 6}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

Factor $2$ out of $2 x$ .

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

Step 3.1.2

Factor $2$ out of $6$ .

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

Step 3.1.3

Factor $2$ out of $2 x + 2 \cdot 3$ .

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

Step 3.2

Simplify terms.

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Step 3.2.1

Cancel the common factor of $x + 3$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

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

Step 3.2.2

Cancel the common factor of $x - 4$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.2.4

Combine $x$ and $\frac{x + 4}{2}$ .

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

Step 3.2.5

Combine $5$ and $\frac{x + 4}{2}$ .

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

Step 3.2.6

Combine the numerators over the common denominator.

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

Step 3.3

Simplify each term.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Multiply $x$ by $x$ .

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

Step 3.3.3

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

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

Step 3.3.4

Apply the distributive property.

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

Step 3.3.5

Multiply $5$ by $4$ .

$\frac{x^{2} + 4 x + 5 x + 20}{2}$

Step 3.4

Add $4 x$ and $5 x$ .

$\frac{x^{2} + 9 x + 20}{2}$

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.

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