Simplify (2x+7)/(x-5)-5/(x^2-10x+25)

$\frac{2 x + 7}{x - 5} - \frac{5}{x^{2} - 10 x + 25}$

Factor using the perfect square rule.

Rewrite $25$ as $5^{2}$ .

$\frac{2 x + 7}{x - 5} - \frac{5}{x^{2} - 10 x + 5^{2}}$

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

$10 x = 2 \cdot x \cdot 5$

Rewrite the polynomial.

$\frac{2 x + 7}{x - 5} - \frac{5}{x^{2} - 2 \cdot x \cdot 5 + 5^{2}}$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{2 x + 7}{x - 5} - \frac{5}{{(x - 5)}^{2}}$

To write $\frac{2 x + 7}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{2 x + 7}{x - 5} \cdot \frac{x - 5}{x - 5} - \frac{5}{{(x - 5)}^{2}}$

Write each expression with a common denominator of ${(x - 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2 x + 7}{x - 5}$ by $\frac{x - 5}{x - 5}$ .

$\frac{(2 x + 7) (x - 5)}{(x - 5) (x - 5)} - \frac{5}{{(x - 5)}^{2}}$

Raise $x - 5$ to the power of $1$ .

$\frac{(2 x + 7) (x - 5)}{{(x - 5)}^{1} (x - 5)} - \frac{5}{{(x - 5)}^{2}}$

Raise $x - 5$ to the power of $1$ .

$\frac{(2 x + 7) (x - 5)}{{(x - 5)}^{1} {(x - 5)}^{1}} - \frac{5}{{(x - 5)}^{2}}$

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

$\frac{(2 x + 7) (x - 5)}{{(x - 5)}^{1 + 1}} - \frac{5}{{(x - 5)}^{2}}$

Add $1$ and $1$ .

$\frac{(2 x + 7) (x - 5)}{{(x - 5)}^{2}} - \frac{5}{{(x - 5)}^{2}}$

Combine the numerators over the common denominator.

$\frac{(2 x + 7) (x - 5) - 5}{{(x - 5)}^{2}}$

Simplify the numerator.

Expand $(2 x + 7) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{2 x (x - 5) + 7 (x - 5) - 5}{{(x - 5)}^{2}}$

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 5 + 7 (x - 5) - 5}{{(x - 5)}^{2}}$

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 5 + 7 x + 7 \cdot - 5 - 5}{{(x - 5)}^{2}}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot - 5 + 7 x + 7 \cdot - 5 - 5}{{(x - 5)}^{2}}$

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 5 + 7 x + 7 \cdot - 5 - 5}{{(x - 5)}^{2}}$

Multiply $- 5$ by $2$ .

$\frac{2 x^{2} - 10 x + 7 x + 7 \cdot - 5 - 5}{{(x - 5)}^{2}}$

Multiply $7$ by $- 5$ .

$\frac{2 x^{2} - 10 x + 7 x - 35 - 5}{{(x - 5)}^{2}}$

Add $- 10 x$ and $7 x$ .

$\frac{2 x^{2} - 3 x - 35 - 5}{{(x - 5)}^{2}}$

Subtract $5$ from $- 35$ .

$\frac{2 x^{2} - 3 x - 40}{{(x - 5)}^{2}}$

Simplify (2x+7)/(x-5)-5/(x^2-10x+25)

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