Algebra Examples

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

Step 1

Factor using the perfect square rule.

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Step 1.1

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

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

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.

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

Step 1.3

Rewrite the polynomial.

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

Step 1.4

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

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

Move $x$ .

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

Step 5.2.1.1.2

Multiply $x$ by $x$ .

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

Step 5.2.1.2

Multiply $- 5$ by $2$ .

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

Step 5.2.1.3

Multiply $7$ by $- 5$ .

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

Step 5.2.2

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

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

Step 5.3

Subtract $5$ from $- 35$ .

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