Algebra Examples

Simplify $\frac{2 x^{3} - 12 x^{2} + 16 x}{x^{2} + 3 x - 28} \div \frac{4 x^{2} - 36 x + 56}{2 x^{2} - 98}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 x^{3} - 12 x^{2} + 16 x}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 2

Simplify the numerator.

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Step 2.1

Factor $2 x$ out of $2 x^{3} - 12 x^{2} + 16 x$ .

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Step 2.1.1

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

$\frac{2 x (x^{2}) - 12 x^{2} + 16 x}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 2.1.2

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

$\frac{2 x (x^{2}) + 2 x (- 6 x) + 16 x}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 2.1.3

Factor $2 x$ out of $16 x$ .

$\frac{2 x (x^{2}) + 2 x (- 6 x) + 2 x (8)}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 2.1.4

Factor $2 x$ out of $2 x (x^{2}) + 2 x (- 6 x)$ .

$\frac{2 x (x^{2} - 6 x) + 2 x (8)}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 2.1.5

Factor $2 x$ out of $2 x (x^{2} - 6 x) + 2 x (8)$ .

$\frac{2 x (x^{2} - 6 x + 8)}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 2.2

Factor $x^{2} - 6 x + 8$ using the AC method.

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Step 2.2.1

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 $8$ and whose sum is $- 6$ .

$- 4, - 2$

Step 2.2.2

Write the factored form using these integers.

$\frac{2 x ((x - 4) (x - 2))}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

$\frac{2 x (x - 4) (x - 2)}{x^{2} + 3 x - 28} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 3

Factor $x^{2} + 3 x - 28$ 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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 3.2

Write the factored form using these integers.

$\frac{2 x (x - 4) (x - 2)}{(x - 4) (x + 7)} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

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.

$\frac{2 x (x - 4) (x - 2)}{(x - 4) (x + 7)} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 4.1.2

Rewrite the expression.

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 x^{2} - 98}{4 x^{2} - 36 x + 56}$

Step 4.2

Cancel the common factor of $2 x^{2} - 98$ and $4 x^{2} - 36 x + 56$ .

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Step 4.2.1

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

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2}) - 98}{4 x^{2} - 36 x + 56}$

Step 4.2.2

Factor $2$ out of $- 98$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 x^{2} + 2 \cdot - 49}{4 x^{2} - 36 x + 56}$

Step 4.2.3

Factor $2$ out of $2 x^{2} + 2 \cdot - 49$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2} - 49)}{4 x^{2} - 36 x + 56}$

Step 4.2.4

Cancel the common factors.

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Step 4.2.4.1

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

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2} - 49)}{2 (2 x^{2}) - 36 x + 56}$

Step 4.2.4.2

Factor $2$ out of $- 36 x$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2} - 49)}{2 (2 x^{2}) + 2 (- 18 x) + 56}$

Step 4.2.4.3

Factor $2$ out of $2 (2 x^{2}) + 2 (- 18 x)$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2} - 49)}{2 (2 x^{2} - 18 x) + 56}$

Step 4.2.4.4

Factor $2$ out of $56$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2} - 49)}{2 (2 x^{2} - 18 x) + 2 (28)}$

Step 4.2.4.5

Factor $2$ out of $2 (2 x^{2} - 18 x) + 2 (28)$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2} - 49)}{2 (2 x^{2} - 18 x + 28)}$

Step 4.2.4.6

Cancel the common factor.

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{2 (x^{2} - 49)}{2 (2 x^{2} - 18 x + 28)}$

Step 4.2.4.7

Rewrite the expression.

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{x^{2} - 49}{2 x^{2} - 18 x + 28}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $49$ as $7^{2}$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{x^{2} - 7^{2}}{2 x^{2} - 18 x + 28}$

Step 5.2

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 $b = 7$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{(x + 7) (x - 7)}{2 x^{2} - 18 x + 28}$

Step 6

Simplify the denominator.

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Step 6.1

Factor $2$ out of $2 x^{2} - 18 x + 28$ .

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Step 6.1.1

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

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{(x + 7) (x - 7)}{2 (x^{2}) - 18 x + 28}$

Step 6.1.2

Factor $2$ out of $- 18 x$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{(x + 7) (x - 7)}{2 (x^{2}) + 2 (- 9 x) + 28}$

Step 6.1.3

Factor $2$ out of $28$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{(x + 7) (x - 7)}{2 x^{2} + 2 (- 9 x) + 2 \cdot 14}$

Step 6.1.4

Factor $2$ out of $2 x^{2} + 2 (- 9 x)$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{(x + 7) (x - 7)}{2 (x^{2} - 9 x) + 2 \cdot 14}$

Step 6.1.5

Factor $2$ out of $2 (x^{2} - 9 x) + 2 \cdot 14$ .

$\frac{2 x (x - 2)}{x + 7} \cdot \frac{(x + 7) (x - 7)}{2 (x^{2} - 9 x + 14)}$

Step 6.2

Factor $x^{2} - 9 x + 14$ using the AC method.

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Step 6.2.1

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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 6.2.2

Write the factored form using these integers.

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

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

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $2 (x - 2)$ .

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Step 7.1.1

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

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

Step 7.1.2

Factor $2 (x - 2)$ out of $2 (x - 7) (x - 2)$ .

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

Step 7.1.3

Cancel the common factor.

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

Step 7.1.4

Rewrite the expression.

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

Step 7.2

Cancel the common factor of $x + 7$ .

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

$x \frac{x - 7}{x - 7}$

Step 7.3

Combine $x$ and $\frac{x - 7}{x - 7}$ .

$\frac{x (x - 7)}{x - 7}$

Step 7.4

Cancel the common factor of $x - 7$ .

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Step 7.4.1

Cancel the common factor.

$\frac{x (x - 7)}{x - 7}$

Step 7.4.2

Divide $x$ by $1$ .

$x$