Finite Math Examples

$\frac{25 - x^{2}}{x^{2} - 5 x}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{5^{2} - x^{2}}{x^{2} - 5 x}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = x$ .

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

Step 2

Simplify terms.

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Step 2.1

Factor $x$ out of $x^{2} - 5 x$ .

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Step 2.1.1

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

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

Step 2.1.2

Factor $x$ out of $- 5 x$ .

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

Step 2.1.3

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

$\frac{(5 + x) (5 - x)}{x (x - 5)}$

Step 2.2

Cancel the common factor of $5 - x$ and $x - 5$ .

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Step 2.2.1

Rewrite $5$ as $- 1 (- 5)$ .

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

Step 2.2.2

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

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

Step 2.2.3

Factor $- 1$ out of $- 1 (- 5) - (x)$ .

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

Step 2.2.4

Reorder terms.

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

Step 2.2.5

Cancel the common factor.

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

Step 2.2.6

Rewrite the expression.

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

Step 2.3

Simplify the expression.

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Step 2.3.1

Move $- 1$ to the left of $5 + x$ .

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

Step 2.3.2

Move the negative in front of the fraction.

$- \frac{5 + x}{x}$