Finite Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Factor $x$ out of $- x$ .

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

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot - 1$ .

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

Step 1.3

Cancel the common factor of $x - 1$ .

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Step 1.3.1

Cancel the common factor.

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

Step 1.3.2

Rewrite the expression.

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

Step 2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3

Simplify terms.

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Step 3.1

Combine $- 1$ and $\frac{x}{x}$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Subtract $x$ from $x$ .

$\frac{0 + 1}{x} (2 x^{2} - x)$

Step 4.2

Add $0$ and $1$ .

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

Step 5

Simplify terms.

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Step 5.1

Multiply $\frac{1}{x}$ by $2 x^{2} - x$ .

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

Step 5.2

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

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Step 5.2.1

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

$\frac{x (2 x) - x}{x}$

Step 5.2.2

Factor $x$ out of $- x$ .

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

Step 5.2.3

Factor $x$ out of $x (2 x) + x \cdot - 1$ .

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

Step 5.3

Cancel the common factor of $x$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Divide $2 x - 1$ by $1$ .

$2 x - 1$