Algebra Examples

$\frac{6 x + 3}{2 x - 3} \div \frac{3 x^{2} - 12 x - 15}{2 x^{2} - x - 3}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 x + 3}{2 x - 3} \cdot \frac{2 x^{2} - x - 3}{3 x^{2} - 12 x - 15}$

Step 2

Factor $3$ out of $6 x + 3$ .

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Step 2.1

Factor $3$ out of $6 x$ .

$\frac{3 (2 x) + 3}{2 x - 3} \cdot \frac{2 x^{2} - x - 3}{3 x^{2} - 12 x - 15}$

Step 2.2

Factor $3$ out of $3$ .

$\frac{3 (2 x) + 3 (1)}{2 x - 3} \cdot \frac{2 x^{2} - x - 3}{3 x^{2} - 12 x - 15}$

Step 2.3

Factor $3$ out of $3 (2 x) + 3 (1)$ .

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{2 x^{2} - x - 3}{3 x^{2} - 12 x - 15}$

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- x$ .

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{2 x^{2} - (x) - 3}{3 x^{2} - 12 x - 15}$

Step 3.1.2

Rewrite $- 1$ as $2$ plus $- 3$

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{2 x^{2} + (2 - 3) x - 3}{3 x^{2} - 12 x - 15}$

Step 3.1.3

Apply the distributive property.

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{2 x^{2} + 2 x - 3 x - 3}{3 x^{2} - 12 x - 15}$

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.

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{(2 x^{2} + 2 x) - 3 x - 3}{3 x^{2} - 12 x - 15}$

Step 3.2.2

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

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{2 x (x + 1) - 3 (x + 1)}{3 x^{2} - 12 x - 15}$

Step 3.3

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

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{(x + 1) (2 x - 3)}{3 x^{2} - 12 x - 15}$

Step 4

Simplify the denominator.

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Step 4.1

Factor $3$ out of $3 x^{2} - 12 x - 15$ .

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Step 4.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (2 x + 1)}{2 x - 3} \cdot \frac{(x + 1) (2 x - 3)}{3 (x^{2}) - 12 x - 15}$

Step 4.1.2

Factor $3$ out of $- 12 x$ .

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

Step 4.1.3

Factor $3$ out of $- 15$ .

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

Step 4.1.4

Factor $3$ out of $3 x^{2} + 3 (- 4 x)$ .

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

Step 4.1.5

Factor $3$ out of $3 (x^{2} - 4 x) + 3 \cdot - 5$ .

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

Step 4.2

Factor $x^{2} - 4 x - 5$ 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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 4.2.2

Write the factored form using these integers.

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

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $3$ .

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Step 5.1.1

Factor $3$ out of $3 (x - 5) (x + 1)$ .

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

Step 5.1.2

Cancel the common factor.

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

Step 5.1.3

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $2 x - 3$ .

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Step 5.2.1

Factor $2 x - 3$ out of $(x + 1) (2 x - 3)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.3

Cancel the common factor of $x + 1$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

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

Step 5.4

Multiply $2 x + 1$ by $\frac{1}{x - 5}$ .

$\frac{2 x + 1}{x - 5}$