Simplify 1/(x^2-14x+49)-(2x)/(x-7)

$\frac{1}{x^{2} - 14 x + 49} - \frac{2 x}{x - 7}$

Factor using the perfect square rule.

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

$\frac{1}{x^{2} - 14 x + 7^{2}} - \frac{2 x}{x - 7}$

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

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

Rewrite the polynomial.

$\frac{1}{x^{2} - 2 \cdot x \cdot 7 + 7^{2}} - \frac{2 x}{x - 7}$

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

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

To write $- \frac{2 x}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

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

Write each expression with a common denominator of ${(x - 7)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 x}{x - 7}$ by $\frac{x - 7}{x - 7}$ .

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

Raise $x - 7$ to the power of $1$ .

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

Raise $x - 7$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{(x - 7)}^{2}} - \frac{2 x (x - 7)}{{(x - 7)}^{1 + 1}}$

Add $1$ and $1$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 7$ by $- 2$ .

$\frac{1 - 2 x^{2} + 14 x}{{(x - 7)}^{2}}$

Reorder terms.

$\frac{- 2 x^{2} + 14 x + 1}{{(x - 7)}^{2}}$

Simplify with factoring out.

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{- (2 x^{2}) + 14 x + 1}{{(x - 7)}^{2}}$

Factor $- 1$ out of $14 x$ .

$\frac{- (2 x^{2}) - (- 14 x) + 1}{{(x - 7)}^{2}}$

Factor $- 1$ out of $- (2 x^{2}) - (- 14 x)$ .

$\frac{- (2 x^{2} - 14 x) + 1}{{(x - 7)}^{2}}$

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

$\frac{- (2 x^{2} - 14 x) - 1 (- 1)}{{(x - 7)}^{2}}$

Factor $- 1$ out of $- (2 x^{2} - 14 x) - 1 (- 1)$ .

$\frac{- (2 x^{2} - 14 x - 1)}{{(x - 7)}^{2}}$

Simplify the expression.

Rewrite $- (2 x^{2} - 14 x - 1)$ as $- 1 (2 x^{2} - 14 x - 1)$ .

$\frac{- 1 (2 x^{2} - 14 x - 1)}{{(x - 7)}^{2}}$

Move the negative in front of the fraction.

$- \frac{2 x^{2} - 14 x - 1}{{(x - 7)}^{2}}$

Simplify 1/(x^2-14x+49)-(2x)/(x-7)

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