Algebra Examples

$\frac{x^{2} - 10 x + 24}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2} - 11 x + 28} \div \frac{3 x^{2} - 147}{x^{2} - 49}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - 10 x + 24}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2} - 11 x + 28} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 2

Factor $x^{2} - 10 x + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 4)}{x^{2} + x - 42} \cdot \frac{x^{2} - 49}{x^{2} - 11 x + 28} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 3

Factor $x^{2} + x - 42$ 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 $- 42$ and whose sum is $1$ .

$- 6, 7$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 4)}{(x - 6) (x + 7)} \cdot \frac{x^{2} - 49}{x^{2} - 11 x + 28} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 4

Simplify the numerator.

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Step 4.1

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

$\frac{(x - 6) (x - 4)}{(x - 6) (x + 7)} \cdot \frac{x^{2} - 7^{2}}{x^{2} - 11 x + 28} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 4.2

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 $b = 7$ .

$\frac{(x - 6) (x - 4)}{(x - 6) (x + 7)} \cdot \frac{(x + 7) (x - 7)}{x^{2} - 11 x + 28} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 5

Factor $x^{2} - 11 x + 28$ using the AC method.

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Step 5.1

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 $- 11$ .

$- 7, - 4$

Step 5.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 4)}{(x - 6) (x + 7)} \cdot \frac{(x + 7) (x - 7)}{(x - 7) (x - 4)} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $x - 4$ .

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Step 6.1.1

Factor $x - 4$ out of $(x - 6) (x - 4)$ .

$\frac{(x - 4) (x - 6)}{(x - 6) (x + 7)} \cdot \frac{(x + 7) (x - 7)}{(x - 7) (x - 4)} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6.1.2

Factor $x - 4$ out of $(x - 7) (x - 4)$ .

$\frac{(x - 4) (x - 6)}{(x - 6) (x + 7)} \cdot \frac{(x + 7) (x - 7)}{(x - 4) (x - 7)} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6.1.3

Cancel the common factor.

$\frac{(x - 4) (x - 6)}{(x - 6) (x + 7)} \cdot \frac{(x + 7) (x - 7)}{(x - 4) (x - 7)} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6.1.4

Rewrite the expression.

$\frac{x - 6}{(x - 6) (x + 7)} \cdot \frac{(x + 7) (x - 7)}{x - 7} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6.2

Cancel the common factor of $x + 7$ .

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Step 6.2.1

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

$\frac{x - 6}{(x + 7) (x - 6)} \cdot \frac{(x + 7) (x - 7)}{x - 7} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6.2.2

Cancel the common factor.

$\frac{x - 6}{(x + 7) (x - 6)} \cdot \frac{(x + 7) (x - 7)}{x - 7} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6.2.3

Rewrite the expression.

$\frac{x - 6}{x - 6} \cdot \frac{x - 7}{x - 7} \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 6.3

Multiply $\frac{x - 6}{x - 6}$ by $\frac{x - 7}{x - 7}$ .

$\frac{(x - 6) (x - 7)}{(x - 6) (x - 7)} \cdot \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 7

Cancel the common factor.

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Step 7.1

Cancel the common factor of $x - 6$ .

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Step 7.1.1

Cancel the common factor.

$\frac{(x - 6) (x - 7)}{(x - 6) (x - 7)} \cdot \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 7.1.2

Rewrite the expression.

$\frac{x - 7}{x - 7} \cdot \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 7.2

Cancel the common factor of $x - 7$ .

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Step 7.2.1

Cancel the common factor.

$\frac{x - 7}{x - 7} \cdot \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 7.2.2

Rewrite the expression.

$1 \frac{x^{2} - 49}{3 x^{2} - 147}$

Step 8

Multiply $\frac{x^{2} - 49}{3 x^{2} - 147}$ by $1$ .

$\frac{x^{2} - 49}{3 x^{2} - 147}$

Step 9

Simplify the numerator.

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Step 9.1

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

$\frac{x^{2} - 7^{2}}{3 x^{2} - 147}$

Step 9.2

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 $b = 7$ .

$\frac{(x + 7) (x - 7)}{3 x^{2} - 147}$

Step 10

Simplify the denominator.

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Step 10.1

Factor $3$ out of $3 x^{2} - 147$ .

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Step 10.1.1

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

$\frac{(x + 7) (x - 7)}{3 (x^{2}) - 147}$

Step 10.1.2

Factor $3$ out of $- 147$ .

$\frac{(x + 7) (x - 7)}{3 x^{2} + 3 \cdot - 49}$

Step 10.1.3

Factor $3$ out of $3 x^{2} + 3 \cdot - 49$ .

$\frac{(x + 7) (x - 7)}{3 (x^{2} - 49)}$

Step 10.2

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

$\frac{(x + 7) (x - 7)}{3 (x^{2} - 7^{2})}$

Step 10.3

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 $b = 7$ .

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

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Cancel the common factor of $x + 7$ .

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Step 11.1.1

Cancel the common factor.

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

Step 11.1.2

Rewrite the expression.

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

Step 11.2

Cancel the common factor of $x - 7$ .

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Step 11.2.1

Cancel the common factor.

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

Step 11.2.2

Rewrite the expression.

$\frac{1}{3}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$