Basic Math Examples

$\frac{x^{2} + 12 x + 24}{x^{2} + 12 x + 27} \cdot \frac{x^{2} + 9 x}{x^{2} + 10 x + 16}$

Step 1

Factor $x^{2} + 12 x + 27$ 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 $27$ and whose sum is $12$ .

$3, 9$

Step 1.2

Write the factored form using these integers.

$\frac{x^{2} + 12 x + 24}{(x + 3) (x + 9)} \cdot \frac{x^{2} + 9 x}{x^{2} + 10 x + 16}$

Step 2

Factor $x$ out of $x^{2} + 9 x$ .

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Step 2.1

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

$\frac{x^{2} + 12 x + 24}{(x + 3) (x + 9)} \cdot \frac{x \cdot x + 9 x}{x^{2} + 10 x + 16}$

Step 2.2

Factor $x$ out of $9 x$ .

$\frac{x^{2} + 12 x + 24}{(x + 3) (x + 9)} \cdot \frac{x \cdot x + x \cdot 9}{x^{2} + 10 x + 16}$

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot 9$ .

$\frac{x^{2} + 12 x + 24}{(x + 3) (x + 9)} \cdot \frac{x (x + 9)}{x^{2} + 10 x + 16}$

Step 3

Factor $x^{2} + 10 x + 16$ 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 $16$ and whose sum is $10$ .

$2, 8$

Step 3.2

Write the factored form using these integers.

$\frac{x^{2} + 12 x + 24}{(x + 3) (x + 9)} \cdot \frac{x (x + 9)}{(x + 2) (x + 8)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 9$ .

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Step 4.1.1

Factor $x + 9$ out of $(x + 3) (x + 9)$ .

$\frac{x^{2} + 12 x + 24}{(x + 9) (x + 3)} \cdot \frac{x (x + 9)}{(x + 2) (x + 8)}$

Step 4.1.2

Factor $x + 9$ out of $x (x + 9)$ .

$\frac{x^{2} + 12 x + 24}{(x + 9) (x + 3)} \cdot \frac{(x + 9) x}{(x + 2) (x + 8)}$

Step 4.1.3

Cancel the common factor.

$\frac{x^{2} + 12 x + 24}{(x + 9) (x + 3)} \cdot \frac{(x + 9) x}{(x + 2) (x + 8)}$

Step 4.1.4

Rewrite the expression.

$\frac{x^{2} + 12 x + 24}{x + 3} \cdot \frac{x}{(x + 2) (x + 8)}$

Step 4.2

Multiply $\frac{x^{2} + 12 x + 24}{x + 3}$ by $\frac{x}{(x + 2) (x + 8)}$ .

$\frac{(x^{2} + 12 x + 24) x}{(x + 3) ((x + 2) (x + 8))}$

Step 4.3

Reorder factors in $\frac{(x^{2} + 12 x + 24) x}{(x + 3) (x + 2) (x + 8)}$ .

$\frac{x (x^{2} + 12 x + 24)}{(x + 3) (x + 2) (x + 8)}$