Calculus Examples

$\frac{\frac{x^{2} - 11 x + 28}{x^{2} - 12 x + 32}}{x^{2} - 15 x + 56}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 11 x + 28}{x^{2} - 12 x + 32} \cdot \frac{1}{x^{2} - 15 x + 56}$

Step 2

Factor $x^{2} - 11 x + 28$ 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 $28$ and whose sum is $- 11$ .

$- 7, - 4$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 7) (x - 4)}{x^{2} - 12 x + 32} \cdot \frac{1}{x^{2} - 15 x + 56}$

Step 3

Factor $x^{2} - 12 x + 32$ 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 $32$ and whose sum is $- 12$ .

$- 8, - 4$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 7) (x - 4)}{(x - 8) (x - 4)} \cdot \frac{1}{x^{2} - 15 x + 56}$

Step 4

Cancel the common factor of $x - 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(x - 7) (x - 4)}{(x - 8) (x - 4)} \cdot \frac{1}{x^{2} - 15 x + 56}$

Step 4.2

Rewrite the expression.

$\frac{x - 7}{x - 8} \cdot \frac{1}{x^{2} - 15 x + 56}$

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 $- 15$ .

$- 8, - 7$

Step 5.2

Write the factored form using these integers.

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 7$ .

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Step 6.1.1

Factor $x - 7$ out of $(x - 8) (x - 7)$ .

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

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

$\frac{1}{x - 8} \cdot \frac{1}{x - 8}$

Step 6.2

Multiply $\frac{1}{x - 8}$ by $\frac{1}{x - 8}$ .

$\frac{1}{(x - 8) (x - 8)}$

Step 7

Raise $x - 8$ to the power of $1$ .

$\frac{1}{{(x - 8)}^{1} (x - 8)}$

Step 8

Raise $x - 8$ to the power of $1$ .

$\frac{1}{{(x - 8)}^{1} {(x - 8)}^{1}}$

Step 9

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

$\frac{1}{{(x - 8)}^{1 + 1}}$

Step 10

Add $1$ and $1$ .

$\frac{1}{{(x - 8)}^{2}}$