Precalculus Examples

$\frac{\frac{x^{2} - 14 x + 49}{x^{2} - 49}}{\frac{3 x - 21}{x + 7}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} - 14 x + 49}{x^{2} - 49} \cdot \frac{x + 7}{3 x - 21}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{x^{2} - 14 x + 7^{2}}{x^{2} - 49} \cdot \frac{x + 7}{3 x - 21}$

Step 2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 2.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 7 + 7^{2}}{x^{2} - 49} \cdot \frac{x + 7}{3 x - 21}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 7$ .

$\frac{{(x - 7)}^{2}}{x^{2} - 49} \cdot \frac{x + 7}{3 x - 21}$

Step 3

Simplify the denominator.

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Step 3.1

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

$\frac{{(x - 7)}^{2}}{x^{2} - 7^{2}} \cdot \frac{x + 7}{3 x - 21}$

Step 3.2

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 $b = 7$ .

$\frac{{(x - 7)}^{2}}{(x + 7) (x - 7)} \cdot \frac{x + 7}{3 x - 21}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 7$ .

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Step 4.1.1

Cancel the common factor.

$\frac{{(x - 7)}^{2}}{(x + 7) (x - 7)} \cdot \frac{x + 7}{3 x - 21}$

Step 4.1.2

Rewrite the expression.

$\frac{{(x - 7)}^{2}}{x - 7} \cdot \frac{1}{3 x - 21}$

Step 4.2

Multiply $\frac{{(x - 7)}^{2}}{x - 7}$ by $\frac{1}{3 x - 21}$ .

$\frac{{(x - 7)}^{2}}{(x - 7) (3 x - 21)}$

Step 4.3

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

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Step 4.3.1

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

$\frac{(x - 7) (x - 7)}{(x - 7) (3 x - 21)}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Cancel the common factor.

$\frac{(x - 7) (x - 7)}{(x - 7) (3 x - 21)}$

Step 4.3.2.2

Rewrite the expression.

$\frac{x - 7}{3 x - 21}$

Step 4.4

Factor $3$ out of $3 x - 21$ .

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Step 4.4.1

Factor $3$ out of $3 x$ .

$\frac{x - 7}{3 (x) - 21}$

Step 4.4.2

Factor $3$ out of $- 21$ .

$\frac{x - 7}{3 x + 3 \cdot - 7}$

Step 4.4.3

Factor $3$ out of $3 x + 3 \cdot - 7$ .

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

Step 4.5

Cancel the common factor of $x - 7$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

$\frac{1}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$