Algebra Examples

$\frac{8 x - 20}{x^{2} + 2 x - 35} \cdot \frac{x^{2} - 7 x + 10}{4 x^{2} - 16}$

Step 1

Factor $4$ out of $8 x - 20$ .

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Step 1.1

Factor $4$ out of $8 x$ .

$\frac{4 (2 x) - 20}{x^{2} + 2 x - 35} \cdot \frac{x^{2} - 7 x + 10}{4 x^{2} - 16}$

Step 1.2

Factor $4$ out of $- 20$ .

$\frac{4 (2 x) + 4 (- 5)}{x^{2} + 2 x - 35} \cdot \frac{x^{2} - 7 x + 10}{4 x^{2} - 16}$

Step 1.3

Factor $4$ out of $4 (2 x) + 4 (- 5)$ .

$\frac{4 (2 x - 5)}{x^{2} + 2 x - 35} \cdot \frac{x^{2} - 7 x + 10}{4 x^{2} - 16}$

Step 2

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

$- 5, 7$

Step 2.2

Write the factored form using these integers.

$\frac{4 (2 x - 5)}{(x - 5) (x + 7)} \cdot \frac{x^{2} - 7 x + 10}{4 x^{2} - 16}$

Step 3

Factor $x^{2} - 7 x + 10$ 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 $10$ and whose sum is $- 7$ .

$- 5, - 2$

Step 3.2

Write the factored form using these integers.

$\frac{4 (2 x - 5)}{(x - 5) (x + 7)} \cdot \frac{(x - 5) (x - 2)}{4 x^{2} - 16}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $4$ out of $4 x^{2} - 16$ .

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Step 4.1.1

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

$\frac{4 (2 x - 5)}{(x - 5) (x + 7)} \cdot \frac{(x - 5) (x - 2)}{4 (x^{2}) - 16}$

Step 4.1.2

Factor $4$ out of $- 16$ .

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

Step 4.1.3

Factor $4$ out of $4 x^{2} + 4 \cdot - 4$ .

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

Step 4.2

Rewrite $4$ as $2^{2}$ .

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $4$ .

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Step 5.1.1

Factor $4$ out of $4 (x + 2) (x - 2)$ .

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $x - 5$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Multiply $\frac{2 x - 5}{x + 7}$ by $\frac{x - 2}{(x + 2) (x - 2)}$ .

$\frac{(2 x - 5) (x - 2)}{(x + 7) ((x + 2) (x - 2))}$

Step 5.4

Cancel the common factor of $x - 2$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(2 x - 5) (x - 2)}{(x + 7) ((x + 2) (x - 2))}$

Step 5.4.2

Rewrite the expression.

$\frac{2 x - 5}{(x + 7) (x + 2)}$