Algebra Examples

$\frac{x^{2} + x - 12}{x^{2} - x - 20} \div \frac{3 x^{2} - 24 x + 45}{12 x^{2} - 48 x - 60}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + x - 12}{x^{2} - x - 20} \cdot \frac{12 x^{2} - 48 x - 60}{3 x^{2} - 24 x + 45}$

Step 2

Factor $x^{2} + x - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 4)}{x^{2} - x - 20} \cdot \frac{12 x^{2} - 48 x - 60}{3 x^{2} - 24 x + 45}$

Step 3

Factor $x^{2} - x - 20$ 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 3) (x + 4)}{(x - 5) (x + 4)} \cdot \frac{12 x^{2} - 48 x - 60}{3 x^{2} - 24 x + 45}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x + 4$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(x - 3) (x + 4)}{(x - 5) (x + 4)} \cdot \frac{12 x^{2} - 48 x - 60}{3 x^{2} - 24 x + 45}$

Step 4.1.2

Rewrite the expression.

$\frac{x - 3}{x - 5} \cdot \frac{12 x^{2} - 48 x - 60}{3 x^{2} - 24 x + 45}$

Step 4.2

Cancel the common factor of $12 x^{2} - 48 x - 60$ and $3 x^{2} - 24 x + 45$ .

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Step 4.2.1

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

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2}) - 48 x - 60}{3 x^{2} - 24 x + 45}$

Step 4.2.2

Factor $3$ out of $- 48 x$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2}) + 3 (- 16 x) - 60}{3 x^{2} - 24 x + 45}$

Step 4.2.3

Factor $3$ out of $3 (4 x^{2}) + 3 (- 16 x)$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x) - 60}{3 x^{2} - 24 x + 45}$

Step 4.2.4

Factor $3$ out of $- 60$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x) + 3 \cdot - 20}{3 x^{2} - 24 x + 45}$

Step 4.2.5

Factor $3$ out of $3 (4 x^{2} - 16 x) + 3 (- 20)$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x - 20)}{3 x^{2} - 24 x + 45}$

Step 4.2.6

Cancel the common factors.

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Step 4.2.6.1

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

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x - 20)}{3 (x^{2}) - 24 x + 45}$

Step 4.2.6.2

Factor $3$ out of $- 24 x$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x - 20)}{3 (x^{2}) + 3 (- 8 x) + 45}$

Step 4.2.6.3

Factor $3$ out of $3 (x^{2}) + 3 (- 8 x)$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x - 20)}{3 (x^{2} - 8 x) + 45}$

Step 4.2.6.4

Factor $3$ out of $45$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x - 20)}{3 (x^{2} - 8 x) + 3 \cdot 15}$

Step 4.2.6.5

Factor $3$ out of $3 (x^{2} - 8 x) + 3 \cdot 15$ .

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x - 20)}{3 (x^{2} - 8 x + 15)}$

Step 4.2.6.6

Cancel the common factor.

$\frac{x - 3}{x - 5} \cdot \frac{3 (4 x^{2} - 16 x - 20)}{3 (x^{2} - 8 x + 15)}$

Step 4.2.6.7

Rewrite the expression.

$\frac{x - 3}{x - 5} \cdot \frac{4 x^{2} - 16 x - 20}{x^{2} - 8 x + 15}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $4$ out of $4 x^{2} - 16 x - 20$ .

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Step 5.1.1

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

$\frac{x - 3}{x - 5} \cdot \frac{4 (x^{2}) - 16 x - 20}{x^{2} - 8 x + 15}$

Step 5.1.2

Factor $4$ out of $- 16 x$ .

$\frac{x - 3}{x - 5} \cdot \frac{4 (x^{2}) + 4 (- 4 x) - 20}{x^{2} - 8 x + 15}$

Step 5.1.3

Factor $4$ out of $- 20$ .

$\frac{x - 3}{x - 5} \cdot \frac{4 x^{2} + 4 (- 4 x) + 4 \cdot - 5}{x^{2} - 8 x + 15}$

Step 5.1.4

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

$\frac{x - 3}{x - 5} \cdot \frac{4 (x^{2} - 4 x) + 4 \cdot - 5}{x^{2} - 8 x + 15}$

Step 5.1.5

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

$\frac{x - 3}{x - 5} \cdot \frac{4 (x^{2} - 4 x - 5)}{x^{2} - 8 x + 15}$

Step 5.2

Factor $x^{2} - 4 x - 5$ using the AC method.

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Step 5.2.1

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 5.2.2

Write the factored form using these integers.

$\frac{x - 3}{x - 5} \cdot \frac{4 ((x - 5) (x + 1))}{x^{2} - 8 x + 15}$

$\frac{x - 3}{x - 5} \cdot \frac{4 (x - 5) (x + 1)}{x^{2} - 8 x + 15}$

Step 6

Factor $x^{2} - 8 x + 15$ using the AC method.

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Step 6.1

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 $- 8$ .

$- 5, - 3$

Step 6.2

Write the factored form using these integers.

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $x - 3$ .

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Step 7.1.1

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

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

Step 7.1.2

Cancel the common factor.

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

Step 7.1.3

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $x - 5$ .

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Step 7.2.1

Factor $x - 5$ out of $4 (x - 5) (x + 1)$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

$\frac{4 (x + 1)}{x - 5}$