Algebra Examples

$\frac{x + 8}{x - 9} \cdot \frac{x^{2} + 10 x + 16}{x - 8} \div \frac{x^{2} + 10 x + 16}{4 x^{2} - 68 x + 288}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 8}{x - 9} \cdot \frac{x^{2} + 10 x + 16}{x - 8} \frac{4 x^{2} - 68 x + 288}{x^{2} + 10 x + 16}$

Step 2

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

$2, 8$

Step 2.2

Write the factored form using these integers.

$\frac{x + 8}{x - 9} \cdot \frac{(x + 2) (x + 8)}{x - 8} \frac{4 x^{2} - 68 x + 288}{x^{2} + 10 x + 16}$

Step 3

Multiply $\frac{x + 8}{x - 9} \cdot \frac{(x + 2) (x + 8)}{x - 8}$ .

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Step 3.1

Multiply $\frac{x + 8}{x - 9}$ by $\frac{(x + 2) (x + 8)}{x - 8}$ .

$\frac{(x + 8) ((x + 2) (x + 8))}{(x - 9) (x - 8)} \cdot \frac{4 x^{2} - 68 x + 288}{x^{2} + 10 x + 16}$

Step 3.2

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

$\frac{{(x + 8)}^{1} (x + 8) (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 x^{2} - 68 x + 288}{x^{2} + 10 x + 16}$

Step 3.3

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

$\frac{{(x + 8)}^{1} {(x + 8)}^{1} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 x^{2} - 68 x + 288}{x^{2} + 10 x + 16}$

Step 3.4

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

$\frac{{(x + 8)}^{1 + 1} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 x^{2} - 68 x + 288}{x^{2} + 10 x + 16}$

Step 3.5

Add $1$ and $1$ .

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 x^{2} - 68 x + 288}{x^{2} + 10 x + 16}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $4$ out of $4 x^{2} - 68 x + 288$ .

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Step 4.1.1

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

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 (x^{2}) - 68 x + 288}{x^{2} + 10 x + 16}$

Step 4.1.2

Factor $4$ out of $- 68 x$ .

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 (x^{2}) + 4 (- 17 x) + 288}{x^{2} + 10 x + 16}$

Step 4.1.3

Factor $4$ out of $288$ .

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 x^{2} + 4 (- 17 x) + 4 \cdot 72}{x^{2} + 10 x + 16}$

Step 4.1.4

Factor $4$ out of $4 x^{2} + 4 (- 17 x)$ .

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 (x^{2} - 17 x) + 4 \cdot 72}{x^{2} + 10 x + 16}$

Step 4.1.5

Factor $4$ out of $4 (x^{2} - 17 x) + 4 \cdot 72$ .

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 (x^{2} - 17 x + 72)}{x^{2} + 10 x + 16}$

Step 4.2

Factor $x^{2} - 17 x + 72$ using the AC method.

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Step 4.2.1

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 $72$ and whose sum is $- 17$ .

$- 9, - 8$

Step 4.2.2

Write the factored form using these integers.

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 ((x - 9) (x - 8))}{x^{2} + 10 x + 16}$

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 (x - 9) (x - 8)}{x^{2} + 10 x + 16}$

Step 5

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

$2, 8$

Step 5.2

Write the factored form using these integers.

$\frac{{(x + 8)}^{2} (x + 2)}{(x - 9) (x - 8)} \cdot \frac{4 (x - 9) (x - 8)}{(x + 2) (x + 8)}$

Step 6

Simplify terms.

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Step 6.1

Combine.

$\frac{{(x + 8)}^{2} (x + 2) (4 (x - 9) (x - 8))}{(x - 9) (x - 8) ((x + 2) (x + 8))}$

Step 6.2

Cancel the common factor of ${(x + 8)}^{2}$ and $x + 8$ .

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Step 6.2.1

Factor $x + 8$ out of ${(x + 8)}^{2} (x + 2) (4 (x - 9) (x - 8))$ .

$\frac{(x + 8) (((x + 8) (x + 2)) (4 (x - 9) (x - 8)))}{(x - 9) (x - 8) ((x + 2) (x + 8))}$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $x + 8$ out of $(x - 9) (x - 8) ((x + 2) (x + 8))$ .

$\frac{(x + 8) (((x + 8) (x + 2)) (4 (x - 9) (x - 8)))}{(x + 8) ((x - 9) (x - 8) (x + 2))}$

Step 6.2.2.2

Cancel the common factor.

$\frac{(x + 8) (((x + 8) (x + 2)) (4 (x - 9) (x - 8)))}{(x + 8) ((x - 9) (x - 8) (x + 2))}$

Step 6.2.2.3

Rewrite the expression.

$\frac{((x + 8) (x + 2)) (4 (x - 9) (x - 8))}{(x - 9) (x - 8) (x + 2)}$

Step 6.3

Cancel the common factor of $x + 2$ .

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Step 6.3.1

Cancel the common factor.

$\frac{(x + 8) (x + 2) (4 (x - 9) (x - 8))}{(x - 9) (x - 8) (x + 2)}$

Step 6.3.2

Rewrite the expression.

$\frac{(x + 8) (4 (x - 9) (x - 8))}{(x - 9) (x - 8)}$

Step 6.4

Cancel the common factor of $x - 9$ .

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Step 6.4.1

Cancel the common factor.

$\frac{(x + 8) (4 (x - 9) (x - 8))}{(x - 9) (x - 8)}$

Step 6.4.2

Rewrite the expression.

$\frac{(x + 8) (4 (x - 8))}{x - 8}$

Step 6.5

Cancel the common factor of $x - 8$ .

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Step 6.5.1

Cancel the common factor.

$\frac{(x + 8) (4 (x - 8))}{x - 8}$

Step 6.5.2

Divide $(x + 8) \cdot (4)$ by $1$ .

$(x + 8) \cdot (4)$

Step 6.6

Apply the distributive property.

$x \cdot 4 + 8 \cdot 4$

Step 6.7

Simplify the expression.

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Step 6.7.1

Move $4$ to the left of $x$ .

$4 \cdot x + 8 \cdot 4$

Step 6.7.2

Multiply $8$ by $4$ .

$4 x + 32$