Algebra Examples

$\frac{2 x - 2}{x + 5} \div \frac{x^{2} - 4 x + 3}{2 x^{2} + 13 x + 15}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 x - 2}{x + 5} \cdot \frac{2 x^{2} + 13 x + 15}{x^{2} - 4 x + 3}$

Step 2

Factor $2$ out of $2 x - 2$ .

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Step 2.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) - 2}{x + 5} \cdot \frac{2 x^{2} + 13 x + 15}{x^{2} - 4 x + 3}$

Step 2.2

Factor $2$ out of $- 2$ .

$\frac{2 (x) + 2 (- 1)}{x + 5} \cdot \frac{2 x^{2} + 13 x + 15}{x^{2} - 4 x + 3}$

Step 2.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$\frac{2 (x - 1)}{x + 5} \cdot \frac{2 x^{2} + 13 x + 15}{x^{2} - 4 x + 3}$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 15 = 30$ and whose sum is $b = 13$ .

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Step 3.1.1

Factor $13$ out of $13 x$ .

$\frac{2 (x - 1)}{x + 5} \cdot \frac{2 x^{2} + 13 (x) + 15}{x^{2} - 4 x + 3}$

Step 3.1.2

Rewrite $13$ as $3$ plus $10$

$\frac{2 (x - 1)}{x + 5} \cdot \frac{2 x^{2} + (3 + 10) x + 15}{x^{2} - 4 x + 3}$

Step 3.1.3

Apply the distributive property.

$\frac{2 (x - 1)}{x + 5} \cdot \frac{2 x^{2} + 3 x + 10 x + 15}{x^{2} - 4 x + 3}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{2 (x - 1)}{x + 5} \cdot \frac{(2 x^{2} + 3 x) + 10 x + 15}{x^{2} - 4 x + 3}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 (x - 1)}{x + 5} \cdot \frac{x (2 x + 3) + 5 (2 x + 3)}{x^{2} - 4 x + 3}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 3$ .

$\frac{2 (x - 1)}{x + 5} \cdot \frac{(2 x + 3) (x + 5)}{x^{2} - 4 x + 3}$

Step 4

Factor $x^{2} - 4 x + 3$ 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 $3$ and whose sum is $- 4$ .

$- 3, - 1$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 1$ .

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Step 5.1.1

Factor $x - 1$ out of $2 (x - 1)$ .

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

Step 5.1.2

Factor $x - 1$ out of $(x - 3) (x - 1)$ .

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

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $x + 5$ .

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Step 5.2.1

Factor $x + 5$ out of $(2 x + 3) (x + 5)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$2 \frac{2 x + 3}{x - 3}$

Step 5.3

Combine $2$ and $\frac{2 x + 3}{x - 3}$ .

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