Finite Math Examples

$\frac{x^{2} - 1}{x - 4} = x + 4$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $x$ from both sides of the equation.

$\frac{x^{2} - 1}{x - 4} - x = 4$

Step 1.2

Subtract $4$ from both sides of the equation.

$\frac{x^{2} - 1}{x - 4} - x - 4 = 0$

Step 2

Simplify $\frac{x^{2} - 1}{x - 4} - x - 4$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

$\frac{x^{2} - 1^{2}}{x - 4} - x - 4 = 0$

Step 2.1.2

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 $b = 1$ .

$\frac{(x + 1) (x - 1)}{x - 4} - x - 4 = 0$

Step 2.2

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

$\frac{(x + 1) (x - 1)}{x - 4} - x \cdot \frac{x - 4}{x - 4} - 4 = 0$

Step 2.3

Combine $- x$ and $\frac{x - 4}{x - 4}$ .

$\frac{(x + 1) (x - 1)}{x - 4} + \frac{- x (x - 4)}{x - 4} - 4 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{(x + 1) (x - 1) - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 2.5.1.1

Apply the distributive property.

$\frac{x (x - 1) + 1 (x - 1) - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 + 1 (x - 1) - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.2

Simplify and combine like terms.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.2.1.2

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

$\frac{x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.2.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} - x + 1 x + 1 \cdot - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.2.1.4

Multiply $x$ by $1$ .

$\frac{x^{2} - x + x + 1 \cdot - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.2.1.5

Multiply $- 1$ by $1$ .

$\frac{x^{2} - x + x - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.2.2

Add $- x$ and $x$ .

$\frac{x^{2} + 0 - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.2.3

Add $x^{2}$ and $0$ .

$\frac{x^{2} - 1 - x (x - 4)}{x - 4} - 4 = 0$

Step 2.5.3

Apply the distributive property.

$\frac{x^{2} - 1 - x \cdot x - x \cdot - 4}{x - 4} - 4 = 0$

Step 2.5.4

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

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Step 2.5.4.1

Move $x$ .

$\frac{x^{2} - 1 - (x \cdot x) - x \cdot - 4}{x - 4} - 4 = 0$

Step 2.5.4.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 1 - x^{2} - x \cdot - 4}{x - 4} - 4 = 0$

Step 2.5.5

Multiply $- 4$ by $- 1$ .

$\frac{x^{2} - 1 - x^{2} + 4 x}{x - 4} - 4 = 0$

Step 2.5.6

Subtract $x^{2}$ from $x^{2}$ .

$\frac{0 - 1 + 4 x}{x - 4} - 4 = 0$

Step 2.5.7

Subtract $1$ from $0$ .

$\frac{- 1 + 4 x}{x - 4} - 4 = 0$

Step 2.6

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

$\frac{- 1 + 4 x}{x - 4} - 4 \cdot \frac{x - 4}{x - 4} = 0$

Step 2.7

Combine $- 4$ and $\frac{x - 4}{x - 4}$ .

$\frac{- 1 + 4 x}{x - 4} + \frac{- 4 (x - 4)}{x - 4} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{- 1 + 4 x - 4 (x - 4)}{x - 4} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{- 1 + 4 x - 4 x - 4 \cdot - 4}{x - 4} = 0$

Step 2.9.2

Multiply $- 4$ by $- 4$ .

$\frac{- 1 + 4 x - 4 x + 16}{x - 4} = 0$

Step 2.9.3

Add $- 1$ and $16$ .

$\frac{4 x - 4 x + 15}{x - 4} = 0$

Step 2.9.4

Subtract $4 x$ from $4 x$ .

$\frac{0 + 15}{x - 4} = 0$

Step 2.9.5

Add $0$ and $15$ .

$\frac{15}{x - 4} = 0$

Step 3

Set the numerator equal to zero.

$15 = 0$

Step 4

Since $15 \neq 0$ , there are no solutions.

No solution