Simplify (x^2-26x+169)/(1859-11x^2)

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

Step 1

Factor using the perfect square rule.

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

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

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

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

Rewrite the polynomial.

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

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.

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

Factor $11$ out of $1859$ .

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

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

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

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

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

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

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

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.

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

Factor $- 1$ out of $x$ .

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

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

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

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

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

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

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

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

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

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

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

Reorder terms.

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

Simplify (x^2-26x+169)/(1859-11x^2)

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