Algebra Examples

$\frac{x^{2} - 26 x + 169}{1859 - 11 x^{2}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $169$ as $13^{2}$ .

$\frac{x^{2} - 26 x + 13^{2}}{1859 - 11 x^{2}}$

Step 1.2

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

$26 x = 2 \cdot x \cdot 13$

Step 1.3

Rewrite the polynomial.

$\frac{x^{2} - 2 \cdot x \cdot 13 + 13^{2}}{1859 - 11 x^{2}}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 13$ .

$\frac{{(x - 13)}^{2}}{1859 - 11 x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $11$ out of $1859 - 11 x^{2}$ .

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Step 2.1.1

Factor $11$ out of $1859$ .

$\frac{{(x - 13)}^{2}}{11 (169) - 11 x^{2}}$

Step 2.1.2

Factor $11$ out of $- 11 x^{2}$ .

$\frac{{(x - 13)}^{2}}{11 (169) + 11 (- x^{2})}$

Step 2.1.3

Factor $11$ out of $11 (169) + 11 (- x^{2})$ .

$\frac{{(x - 13)}^{2}}{11 (169 - x^{2})}$

Step 2.2

Rewrite $169$ as $13^{2}$ .

$\frac{{(x - 13)}^{2}}{11 (13^{2} - x^{2})}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 13$ and $b = x$ .

$\frac{{(x - 13)}^{2}}{11 (13 + x) (13 - x)}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of ${(x - 13)}^{2}$ and $13 - x$ .

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Step 3.1.1

Factor $- 1$ out of $x$ .

$\frac{{(- 1 (- x) - 13)}^{2}}{11 (13 + x) (13 - x)}$

Step 3.1.2

Rewrite $- 13$ as $- 1 (13)$ .

$\frac{{(- 1 (- x) - 1 (13))}^{2}}{11 (13 + x) (13 - x)}$

Step 3.1.3

Factor $- 1$ out of $- 1 (- x) - 1 (13)$ .

$\frac{{(- 1 (- x + 13))}^{2}}{11 (13 + x) (13 - x)}$

Step 3.1.4

Apply the product rule to $- 1 (- x + 13)$ .

$\frac{{(- 1)}^{2} {(- x + 13)}^{2}}{11 (13 + x) (13 - x)}$

Step 3.1.5

Raise $- 1$ to the power of $2$ .

$\frac{1 {(- x + 13)}^{2}}{11 (13 + x) (13 - x)}$

Step 3.1.6

Multiply ${(- x + 13)}^{2}$ by $1$ .

$\frac{{(- x + 13)}^{2}}{11 (13 + x) (13 - x)}$

Step 3.1.7

Reorder terms.

$\frac{{(- x + 13)}^{2}}{11 (13 + x) (- x + 13)}$

Step 3.1.8

Factor $- x + 13$ out of ${(- x + 13)}^{2}$ .

$\frac{(- x + 13) (- x + 13)}{11 (13 + x) (- x + 13)}$

Step 3.1.9

Cancel the common factors.

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Step 3.1.9.1

Factor $- x + 13$ out of $11 (13 + x) (- x + 13)$ .

$\frac{(- x + 13) (- x + 13)}{(- x + 13) (11 (13 + x))}$

Step 3.1.9.2

Cancel the common factor.

$\frac{(- x + 13) (- x + 13)}{(- x + 13) (11 (13 + x))}$

Step 3.1.9.3

Rewrite the expression.

$\frac{- x + 13}{11 (13 + x)}$

Step 3.2

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

$\frac{- (x) + 13}{11 (13 + x)}$

Step 3.3

Rewrite $13$ as $- 1 (- 13)$ .

$\frac{- (x) - 1 (- 13)}{11 (13 + x)}$

Step 3.4

Factor $- 1$ out of $- (x) - 1 (- 13)$ .

$\frac{- (x - 13)}{11 (13 + x)}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Rewrite $- (x - 13)$ as $- 1 (x - 13)$ .

$\frac{- 1 (x - 13)}{11 (13 + x)}$

Step 3.5.2

Move the negative in front of the fraction.

$- \frac{x - 13}{11 (13 + x)}$