Algebra Examples

$\frac{x^{2} + 15 x + 56}{x} \div \frac{x^{2} + 2 x - 35}{4 x - 20}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 15 x + 56}{x} \cdot \frac{4 x - 20}{x^{2} + 2 x - 35}$

Step 2

Factor $x^{2} + 15 x + 56$ 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 $56$ and whose sum is $15$ .

$7, 8$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 7) (x + 8)}{x} \cdot \frac{4 x - 20}{x^{2} + 2 x - 35}$

Step 3

Factor $4$ out of $4 x - 20$ .

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Step 3.1

Factor $4$ out of $4 x$ .

$\frac{(x + 7) (x + 8)}{x} \cdot \frac{4 (x) - 20}{x^{2} + 2 x - 35}$

Step 3.2

Factor $4$ out of $- 20$ .

$\frac{(x + 7) (x + 8)}{x} \cdot \frac{4 x + 4 \cdot - 5}{x^{2} + 2 x - 35}$

Step 3.3

Factor $4$ out of $4 x + 4 \cdot - 5$ .

$\frac{(x + 7) (x + 8)}{x} \cdot \frac{4 (x - 5)}{x^{2} + 2 x - 35}$

Step 4

Factor $x^{2} + 2 x - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 7) (x + 8)}{x} \cdot \frac{4 (x - 5)}{(x - 5) (x + 7)}$

Step 5

Cancel the common factor of $x + 7$ .

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Step 5.1

Factor $x + 7$ out of $(x - 5) (x + 7)$ .

$\frac{(x + 7) (x + 8)}{x} \cdot \frac{4 (x - 5)}{(x + 7) (x - 5)}$

Step 5.2

Cancel the common factor.

$\frac{(x + 7) (x + 8)}{x} \cdot \frac{4 (x - 5)}{(x + 7) (x - 5)}$

Step 5.3

Rewrite the expression.

$\frac{x + 8}{x} \cdot \frac{4 (x - 5)}{x - 5}$

Step 6

Multiply $\frac{x + 8}{x}$ by $\frac{4 (x - 5)}{x - 5}$ .

$\frac{(x + 8) (4 (x - 5))}{x (x - 5)}$

Step 7

Cancel the common factor of $x - 5$ .

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Step 7.1

Cancel the common factor.

$\frac{(x + 8) (4 (x - 5))}{x (x - 5)}$

Step 7.2

Rewrite the expression.

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

Step 8

Move $4$ to the left of $x + 8$ .

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

Step 9

Apply the distributive property.

$\frac{4 x + 4 \cdot 8}{x}$

Step 10

Multiply $4$ by $8$ .

$\frac{4 x + 32}{x}$

Step 11

Split the fraction $\frac{4 x + 32}{x}$ into two fractions.

$\frac{4 x}{x} + \frac{32}{x}$

Step 12

Cancel the common factor of $x$ .

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Step 12.1

Cancel the common factor.

$\frac{4 x}{x} + \frac{32}{x}$

Step 12.2

Divide $4$ by $1$ .

$4 + \frac{32}{x}$