Algebra Examples

$\frac{2 x^{2} + 7 x - 4}{x^{2} + 8 x + 15} \div \frac{x + 4}{x^{2} + 10 x + 25}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 x^{2} + 7 x - 4}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot - 4 = - 8$ and whose sum is $b = 7$ .

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Step 2.1.1

Factor $7$ out of $7 x$ .

$\frac{2 x^{2} + 7 (x) - 4}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 2.1.2

Rewrite $7$ as $- 1$ plus $8$

$\frac{2 x^{2} + (- 1 + 8) x - 4}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 2.1.3

Apply the distributive property.

$\frac{2 x^{2} - 1 x + 8 x - 4}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 1 x) + 8 x - 4}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 2.2.2

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

$\frac{x (2 x - 1) + 4 (2 x - 1)}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{(2 x - 1) (x + 4)}{x^{2} + 8 x + 15} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 3

Factor $x^{2} + 8 x + 15$ 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 $15$ and whose sum is $8$ .

$3, 5$

Step 3.2

Write the factored form using these integers.

$\frac{(2 x - 1) (x + 4)}{(x + 3) (x + 5)} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 4$ .

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Step 4.1.1

Factor $x + 4$ out of $(2 x - 1) (x + 4)$ .

$\frac{(x + 4) (2 x - 1)}{(x + 3) (x + 5)} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 4.1.2

Cancel the common factor.

$\frac{(x + 4) (2 x - 1)}{(x + 3) (x + 5)} \cdot \frac{x^{2} + 10 x + 25}{x + 4}$

Step 4.1.3

Rewrite the expression.

$\frac{2 x - 1}{(x + 3) (x + 5)} (x^{2} + 10 x + 25)$

Step 4.2

Multiply $\frac{2 x - 1}{(x + 3) (x + 5)}$ by $x^{2} + 10 x + 25$ .

$\frac{(2 x - 1) (x^{2} + 10 x + 25)}{(x + 3) (x + 5)}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $25$ as $5^{2}$ .

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

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 5.3

Rewrite the polynomial.

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

Step 5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

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

Step 6

Cancel the common factor of ${(x + 5)}^{2}$ and $x + 5$ .

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Step 6.1

Factor $x + 5$ out of $(2 x - 1) {(x + 5)}^{2}$ .

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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