Precalculus Examples

$\frac{x^{2} + 9 x + 20}{x^{2} + 11 x + 30} \cdot \frac{x^{2} + 4 x}{x^{2} + 10 x + 28}$

Step 1

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

$4, 5$

Step 1.2

Write the factored form using these integers.

$\frac{(x + 4) (x + 5)}{x^{2} + 11 x + 30} \cdot \frac{x^{2} + 4 x}{x^{2} + 10 x + 28}$

Step 2

Factor $x^{2} + 11 x + 30$ 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 $30$ and whose sum is $11$ .

$5, 6$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 4) (x + 5)}{(x + 5) (x + 6)} \cdot \frac{x^{2} + 4 x}{x^{2} + 10 x + 28}$

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

$\frac{(x + 4) (x + 5)}{(x + 5) (x + 6)} \cdot \frac{x \cdot x + 4 x}{x^{2} + 10 x + 28}$

Step 3.1.2

Factor $x$ out of $4 x$ .

$\frac{(x + 4) (x + 5)}{(x + 5) (x + 6)} \cdot \frac{x \cdot x + x \cdot 4}{x^{2} + 10 x + 28}$

Step 3.1.3

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

$\frac{(x + 4) (x + 5)}{(x + 5) (x + 6)} \cdot \frac{x (x + 4)}{x^{2} + 10 x + 28}$

Step 3.2

Cancel the common factor of $x + 5$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(x + 4) (x + 5)}{(x + 5) (x + 6)} \cdot \frac{x (x + 4)}{x^{2} + 10 x + 28}$

Step 3.2.2

Rewrite the expression.

$\frac{x + 4}{x + 6} \cdot \frac{x (x + 4)}{x^{2} + 10 x + 28}$

Step 4

Multiply $\frac{x + 4}{x + 6} \cdot \frac{x (x + 4)}{x^{2} + 10 x + 28}$ .

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Step 4.1

Multiply $\frac{x + 4}{x + 6}$ by $\frac{x (x + 4)}{x^{2} + 10 x + 28}$ .

$\frac{(x + 4) (x (x + 4))}{(x + 6) (x^{2} + 10 x + 28)}$

Step 4.2

Raise $x + 4$ to the power of $1$ .

$\frac{x ({(x + 4)}^{1} (x + 4))}{(x + 6) (x^{2} + 10 x + 28)}$

Step 4.3

Raise $x + 4$ to the power of $1$ .

$\frac{x ({(x + 4)}^{1} {(x + 4)}^{1})}{(x + 6) (x^{2} + 10 x + 28)}$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x {(x + 4)}^{1 + 1}}{(x + 6) (x^{2} + 10 x + 28)}$

Step 4.5

Add $1$ and $1$ .

$\frac{x {(x + 4)}^{2}}{(x + 6) (x^{2} + 10 x + 28)}$