Algebra Examples

$\frac{6 - x}{- 9} \cdot \frac{- 2 x}{x^{2} - 13 x + 42} \div \frac{6 x^{2} + 42 x}{x^{2} - 49}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 - x}{- 9} \cdot \frac{- 2 x}{x^{2} - 13 x + 42} \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 2

Factor $x^{2} - 13 x + 42$ 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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Step 2.2

Write the factored form using these integers.

$\frac{6 - x}{- 9} \cdot \frac{- 2 x}{(x - 7) (x - 6)} \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 3

Simplify the expression.

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Step 3.1

Move the negative in front of the fraction.

$- \frac{6 - x}{9} \cdot \frac{- 2 x}{(x - 7) (x - 6)} \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{6 - x}{9} \cdot (- \frac{2 x}{(x - 7) (x - 6)}) \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 4

Multiply $- \frac{6 - x}{9} (- \frac{2 x}{(x - 7) (x - 6)})$ .

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Step 4.1

Multiply $- 1$ by $- 1$ .

$1 \frac{6 - x}{9} \frac{2 x}{(x - 7) (x - 6)} \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 4.2

Multiply $\frac{6 - x}{9}$ by $1$ .

$\frac{6 - x}{9} \cdot \frac{2 x}{(x - 7) (x - 6)} \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 4.3

Multiply $\frac{6 - x}{9}$ by $\frac{2 x}{(x - 7) (x - 6)}$ .

$\frac{(6 - x) (2 x)}{9 ((x - 7) (x - 6))} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $6 - x$ and $x - 6$ .

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Step 5.1.1

Rewrite $6$ as $- 1 (- 6)$ .

$\frac{(- 1 (- 6) - x) (2 x)}{9 ((x - 7) (x - 6))} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5.1.2

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

$\frac{(- 1 (- 6) - (x)) (2 x)}{9 ((x - 7) (x - 6))} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5.1.3

Factor $- 1$ out of $- 1 (- 6) - (x)$ .

$\frac{- 1 (- 6 + x) (2 x)}{9 ((x - 7) (x - 6))} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5.1.4

Reorder terms.

$\frac{- 1 (x - 6) (2 x)}{9 (x - 7) (x - 6)} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5.1.5

Cancel the common factor.

$\frac{- 1 (x - 6) (2 x)}{9 (x - 7) (x - 6)} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5.1.6

Rewrite the expression.

$\frac{- 1 (2 x)}{9 (x - 7)} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5.2

Simplify the expression.

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Step 5.2.1

Multiply $- 1$ by $2$ .

$\frac{- 2 x}{9 (x - 7)} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{2 x}{9 (x - 7)} \cdot \frac{x^{2} - 49}{6 x^{2} + 42 x}$

Step 6

Simplify the numerator.

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Step 6.1

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

$- \frac{2 x}{9 (x - 7)} \cdot \frac{x^{2} - 7^{2}}{6 x^{2} + 42 x}$

Step 6.2

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 $b = 7$ .

$- \frac{2 x}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{6 x^{2} + 42 x}$

Step 7

Simplify terms.

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Step 7.1

Factor $6 x$ out of $6 x^{2} + 42 x$ .

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Step 7.1.1

Factor $6 x$ out of $6 x^{2}$ .

$- \frac{2 x}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{6 x (x) + 42 x}$

Step 7.1.2

Factor $6 x$ out of $42 x$ .

$- \frac{2 x}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{6 x (x) + 6 x (7)}$

Step 7.1.3

Factor $6 x$ out of $6 x (x) + 6 x (7)$ .

$- \frac{2 x}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{6 x (x + 7)}$

Step 7.2

Cancel the common factor of $2 x$ .

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Step 7.2.1

Move the leading negative in $- \frac{2 x}{9 (x - 7)}$ into the numerator.

$\frac{- 2 x}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{6 x (x + 7)}$

Step 7.2.2

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

$\frac{2 x (- 1)}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{6 x (x + 7)}$

Step 7.2.3

Factor $2 x$ out of $6 x (x + 7)$ .

$\frac{2 x (- 1)}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{2 x (3 (x + 7))}$

Step 7.2.4

Cancel the common factor.

$\frac{2 x \cdot - 1}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{2 x (3 (x + 7))}$

Step 7.2.5

Rewrite the expression.

$\frac{- 1}{9 (x - 7)} \cdot \frac{(x + 7) (x - 7)}{3 (x + 7)}$

Step 7.3

Cancel the common factor of $x - 7$ .

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Step 7.3.1

Factor $x - 7$ out of $9 (x - 7)$ .

$\frac{- 1}{(x - 7) \cdot 9} \cdot \frac{(x + 7) (x - 7)}{3 (x + 7)}$

Step 7.3.2

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

$\frac{- 1}{(x - 7) \cdot 9} \cdot \frac{(x - 7) (x + 7)}{3 (x + 7)}$

Step 7.3.3

Cancel the common factor.

$\frac{- 1}{(x - 7) \cdot 9} \cdot \frac{(x - 7) (x + 7)}{3 (x + 7)}$

Step 7.3.4

Rewrite the expression.

$\frac{- 1}{9} \cdot \frac{x + 7}{3 (x + 7)}$

Step 7.4

Multiply $\frac{- 1}{9}$ by $\frac{x + 7}{3 (x + 7)}$ .

$\frac{- (x + 7)}{9 (3 (x + 7))}$

Step 7.5

Multiply $3$ by $9$ .

$\frac{- (x + 7)}{27 (x + 7)}$

Step 7.6

Cancel the common factor of $x + 7$ .

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Step 7.6.1

Cancel the common factor.

$\frac{- (x + 7)}{27 (x + 7)}$

Step 7.6.2

Rewrite the expression.

$\frac{- 1}{27}$

Step 7.7

Move the negative in front of the fraction.

$- \frac{1}{27}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{27}$

Decimal Form:

$- 0.\overline{037}$