Pre-Algebra Examples

$\frac{\frac{x^{2} + 14 x + 45}{x^{2} + 16 x + 63}}{\frac{x^{2} + 5 x}{x^{2} + 15 x + 56}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 14 x + 45}{x^{2} + 16 x + 63} \cdot \frac{x^{2} + 15 x + 56}{x^{2} + 5 x}$

Step 2

Factor $x^{2} + 14 x + 45$ 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 $45$ and whose sum is $14$ .

$5, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 5) (x + 9)}{x^{2} + 16 x + 63} \cdot \frac{x^{2} + 15 x + 56}{x^{2} + 5 x}$

Step 3

Factor $x^{2} + 16 x + 63$ 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 $63$ and whose sum is $16$ .

$7, 9$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 5) (x + 9)}{(x + 7) (x + 9)} \cdot \frac{x^{2} + 15 x + 56}{x^{2} + 5 x}$

Step 4

Cancel the common factor of $x + 9$ .

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Step 4.1

Cancel the common factor.

$\frac{(x + 5) (x + 9)}{(x + 7) (x + 9)} \cdot \frac{x^{2} + 15 x + 56}{x^{2} + 5 x}$

Step 4.2

Rewrite the expression.

$\frac{x + 5}{x + 7} \cdot \frac{x^{2} + 15 x + 56}{x^{2} + 5 x}$

Step 5

Factor $x^{2} + 15 x + 56$ using the AC method.

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Step 5.1

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 $56$ and whose sum is $15$ .

$7, 8$

Step 5.2

Write the factored form using these integers.

$\frac{x + 5}{x + 7} \cdot \frac{(x + 7) (x + 8)}{x^{2} + 5 x}$

Step 6

Factor $x$ out of $x^{2} + 5 x$ .

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Step 6.1

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

$\frac{x + 5}{x + 7} \cdot \frac{(x + 7) (x + 8)}{x \cdot x + 5 x}$

Step 6.2

Factor $x$ out of $5 x$ .

$\frac{x + 5}{x + 7} \cdot \frac{(x + 7) (x + 8)}{x \cdot x + x \cdot 5}$

Step 6.3

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

$\frac{x + 5}{x + 7} \cdot \frac{(x + 7) (x + 8)}{x (x + 5)}$

Step 7

Cancel the common factor of $x + 5$ .

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Step 7.1

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

$\frac{x + 5}{x + 7} \cdot \frac{(x + 7) (x + 8)}{(x + 5) x}$

Step 7.2

Cancel the common factor.

$\frac{x + 5}{x + 7} \cdot \frac{(x + 7) (x + 8)}{(x + 5) x}$

Step 7.3

Rewrite the expression.

$\frac{1}{x + 7} \cdot \frac{(x + 7) (x + 8)}{x}$

Step 8

Cancel the common factor of $x + 7$ .

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Step 8.1

Cancel the common factor.

$\frac{1}{x + 7} \cdot \frac{(x + 7) (x + 8)}{x}$

Step 8.2

Rewrite the expression.

$\frac{x + 8}{x}$

Step 9

Split the fraction $\frac{x + 8}{x}$ into two fractions.

$\frac{x}{x} + \frac{8}{x}$

Step 10

Cancel the common factor of $x$ .

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Step 10.1

Cancel the common factor.

$\frac{x}{x} + \frac{8}{x}$

Step 10.2

Rewrite the expression.

$1 + \frac{8}{x}$