Pre-Algebra Examples

$\frac{\frac{\frac{x^{2} + 5 x + 6}{x^{2} - 8 x + 15}}{x + 2}}{x - 5}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{x^{2} + 5 x + 6}{x^{2} - 8 x + 15}}{x + 2} \cdot \frac{1}{x - 5}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 5 x + 6}{x^{2} - 8 x + 15} \cdot \frac{1}{x + 2} \frac{1}{x - 5}$

Step 3

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 3)}{x^{2} - 8 x + 15} \cdot \frac{1}{x + 2} \frac{1}{x - 5}$

Step 4

Factor $x^{2} - 8 x + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 3)}{(x - 5) (x - 3)} \cdot \frac{1}{x + 2} \frac{1}{x - 5}$

Step 5

Cancel the common factor of $x + 2$ .

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Step 5.1

Cancel the common factor.

$\frac{(x + 2) (x + 3)}{(x - 5) (x - 3)} \cdot \frac{1}{x + 2} \frac{1}{x - 5}$

Step 5.2

Rewrite the expression.

$\frac{x + 3}{(x - 5) (x - 3)} \cdot \frac{1}{x - 5}$

Step 6

Multiply $\frac{x + 3}{(x - 5) (x - 3)} \cdot \frac{1}{x - 5}$ .

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Step 6.1

Multiply $\frac{x + 3}{(x - 5) (x - 3)}$ by $\frac{1}{x - 5}$ .

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

Step 6.2

Raise $x - 5$ to the power of $1$ .

$\frac{x + 3}{{(x - 5)}^{1} (x - 5) (x - 3)}$

Step 6.3

Raise $x - 5$ to the power of $1$ .

$\frac{x + 3}{{(x - 5)}^{1} {(x - 5)}^{1} (x - 3)}$

Step 6.4

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

$\frac{x + 3}{{(x - 5)}^{1 + 1} (x - 3)}$

Step 6.5

Add $1$ and $1$ .

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

Step 7

Split the fraction $\frac{x + 3}{{(x - 5)}^{2} (x - 3)}$ into two fractions.

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