Precalculus Examples

$\frac{x}{x^{2} - 10 x + 25} - \frac{x - 4}{2 x - 10}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

$\frac{x}{x^{2} - 10 x + 5^{2}} - \frac{x - 4}{2 x - 10}$

Step 1.1.2

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 1.1.3

Rewrite the polynomial.

$\frac{x}{x^{2} - 2 \cdot x \cdot 5 + 5^{2}} - \frac{x - 4}{2 x - 10}$

Step 1.1.4

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

$\frac{x}{{(x - 5)}^{2}} - \frac{x - 4}{2 x - 10}$

Step 1.2

Factor $2$ out of $2 x - 10$ .

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Step 1.2.1

Factor $2$ out of $2 x$ .

$\frac{x}{{(x - 5)}^{2}} - \frac{x - 4}{2 (x) - 10}$

Step 1.2.2

Factor $2$ out of $- 10$ .

$\frac{x}{{(x - 5)}^{2}} - \frac{x - 4}{2 x + 2 \cdot - 5}$

Step 1.2.3

Factor $2$ out of $2 x + 2 \cdot - 5$ .

$\frac{x}{{(x - 5)}^{2}} - \frac{x - 4}{2 (x - 5)}$

Step 2

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

$\frac{x}{{(x - 5)}^{2}} \cdot \frac{2}{2} - \frac{x - 4}{2 (x - 5)}$

Step 3

To write $- \frac{x - 4}{2 (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{x}{{(x - 5)}^{2}} \cdot \frac{2}{2} - \frac{x - 4}{2 (x - 5)} \cdot \frac{x - 5}{x - 5}$

Step 4

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

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Step 4.1

Multiply $\frac{x}{{(x - 5)}^{2}}$ by $\frac{2}{2}$ .

$\frac{x \cdot 2}{{(x - 5)}^{2} \cdot 2} - \frac{x - 4}{2 (x - 5)} \cdot \frac{x - 5}{x - 5}$

Step 4.2

Multiply $\frac{x - 4}{2 (x - 5)}$ by $\frac{x - 5}{x - 5}$ .

$\frac{x \cdot 2}{{(x - 5)}^{2} \cdot 2} - \frac{(x - 4) (x - 5)}{2 (x - 5) (x - 5)}$

Step 4.3

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

$\frac{x \cdot 2}{{(x - 5)}^{2} \cdot 2} - \frac{(x - 4) (x - 5)}{2 ({(x - 5)}^{1} (x - 5))}$

Step 4.4

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

$\frac{x \cdot 2}{{(x - 5)}^{2} \cdot 2} - \frac{(x - 4) (x - 5)}{2 ({(x - 5)}^{1} {(x - 5)}^{1})}$

Step 4.5

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

$\frac{x \cdot 2}{{(x - 5)}^{2} \cdot 2} - \frac{(x - 4) (x - 5)}{2 {(x - 5)}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{x \cdot 2}{{(x - 5)}^{2} \cdot 2} - \frac{(x - 4) (x - 5)}{2 {(x - 5)}^{2}}$

Step 4.7

Reorder the factors of ${(x - 5)}^{2} \cdot 2$ .

$\frac{x \cdot 2}{2 {(x - 5)}^{2}} - \frac{(x - 4) (x - 5)}{2 {(x - 5)}^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - (x - 4) (x - 5)}{2 {(x - 5)}^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Move $2$ to the left of $x$ .

$\frac{2 \cdot x - (x - 4) (x - 5)}{2 {(x - 5)}^{2}}$

Step 6.2

Apply the distributive property.

$\frac{2 x + (- x - - 4) (x - 5)}{2 {(x - 5)}^{2}}$

Step 6.3

Multiply $- 1$ by $- 4$ .

$\frac{2 x + (- x + 4) (x - 5)}{2 {(x - 5)}^{2}}$

Step 6.4

Expand $(- x + 4) (x - 5)$ using the FOIL Method.

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Step 6.4.1

Apply the distributive property.

$\frac{2 x - x (x - 5) + 4 (x - 5)}{2 {(x - 5)}^{2}}$

Step 6.4.2

Apply the distributive property.

$\frac{2 x - x \cdot x - x \cdot - 5 + 4 (x - 5)}{2 {(x - 5)}^{2}}$

Step 6.4.3

Apply the distributive property.

$\frac{2 x - x \cdot x - x \cdot - 5 + 4 x + 4 \cdot - 5}{2 {(x - 5)}^{2}}$

Step 6.5

Simplify and combine like terms.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

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

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Step 6.5.1.1.1

Move $x$ .

$\frac{2 x - (x \cdot x) - x \cdot - 5 + 4 x + 4 \cdot - 5}{2 {(x - 5)}^{2}}$

Step 6.5.1.1.2

Multiply $x$ by $x$ .

$\frac{2 x - x^{2} - x \cdot - 5 + 4 x + 4 \cdot - 5}{2 {(x - 5)}^{2}}$

Step 6.5.1.2

Multiply $- 5$ by $- 1$ .

$\frac{2 x - x^{2} + 5 x + 4 x + 4 \cdot - 5}{2 {(x - 5)}^{2}}$

Step 6.5.1.3

Multiply $4$ by $- 5$ .

$\frac{2 x - x^{2} + 5 x + 4 x - 20}{2 {(x - 5)}^{2}}$

Step 6.5.2

Add $5 x$ and $4 x$ .

$\frac{2 x - x^{2} + 9 x - 20}{2 {(x - 5)}^{2}}$

Step 6.6

Add $2 x$ and $9 x$ .

$\frac{- x^{2} + 11 x - 20}{2 {(x - 5)}^{2}}$

Step 7

Simplify with factoring out.

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Step 7.1

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

$\frac{- (x^{2}) + 11 x - 20}{2 {(x - 5)}^{2}}$

Step 7.2

Factor $- 1$ out of $11 x$ .

$\frac{- (x^{2}) - (- 11 x) - 20}{2 {(x - 5)}^{2}}$

Step 7.3

Factor $- 1$ out of $- (x^{2}) - (- 11 x)$ .

$\frac{- (x^{2} - 11 x) - 20}{2 {(x - 5)}^{2}}$

Step 7.4

Rewrite $- 20$ as $- 1 (20)$ .

$\frac{- (x^{2} - 11 x) - 1 (20)}{2 {(x - 5)}^{2}}$

Step 7.5

Factor $- 1$ out of $- (x^{2} - 11 x) - 1 (20)$ .

$\frac{- (x^{2} - 11 x + 20)}{2 {(x - 5)}^{2}}$

Step 7.6

Simplify the expression.

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Step 7.6.1

Rewrite $- (x^{2} - 11 x + 20)$ as $- 1 (x^{2} - 11 x + 20)$ .

$\frac{- 1 (x^{2} - 11 x + 20)}{2 {(x - 5)}^{2}}$

Step 7.6.2

Move the negative in front of the fraction.

$- \frac{x^{2} - 11 x + 20}{2 {(x - 5)}^{2}}$