Algebra Examples

$\frac{x^{2} - 49}{6 x - 42} \cdot \frac{6 x^{2} - 294}{x^{2} {(x + 7)}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 7^{2}}{6 x - 42} \cdot \frac{6 x^{2} - 294}{x^{2} {(x + 7)}^{2}}$

Step 1.2

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 $b = 7$ .

$\frac{(x + 7) (x - 7)}{6 x - 42} \cdot \frac{6 x^{2} - 294}{x^{2} {(x + 7)}^{2}}$

Step 2

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

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Step 2.1

Factor $6$ out of $6 x$ .

$\frac{(x + 7) (x - 7)}{6 (x) - 42} \cdot \frac{6 x^{2} - 294}{x^{2} {(x + 7)}^{2}}$

Step 2.2

Factor $6$ out of $- 42$ .

$\frac{(x + 7) (x - 7)}{6 x + 6 \cdot - 7} \cdot \frac{6 x^{2} - 294}{x^{2} {(x + 7)}^{2}}$

Step 2.3

Factor $6$ out of $6 x + 6 \cdot - 7$ .

$\frac{(x + 7) (x - 7)}{6 (x - 7)} \cdot \frac{6 x^{2} - 294}{x^{2} {(x + 7)}^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

Factor $6$ out of $6 x^{2} - 294$ .

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Step 3.1.1

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

$\frac{(x + 7) (x - 7)}{6 (x - 7)} \cdot \frac{6 (x^{2}) - 294}{x^{2} {(x + 7)}^{2}}$

Step 3.1.2

Factor $6$ out of $- 294$ .

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

Step 3.1.3

Factor $6$ out of $6 x^{2} + 6 \cdot - 49$ .

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

Step 3.2

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

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

Step 3.3

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 $b = 7$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $x + 7$ and ${(x + 7)}^{2}$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $x + 7$ out of $6 (x - 7) (x^{2} {(x + 7)}^{2})$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $x - 7$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $6$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

Rewrite the expression.

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

Step 4.5

Cancel the common factor of $x + 7$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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