Algebra Examples

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

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x + 1, 1, 1$

Step 1.2

Remove parentheses.

$x + 1, 1, 1$

Step 1.3

The LCM of one and any expression is the expression.

$x + 1$

Step 2

Multiply each term in $\frac{x}{x + 1} = x - 1$ by $x + 1$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x}{x + 1} = x - 1$ by $x + 1$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $x + 1$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

$x = x (x + 1) - (x + 1)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Apply the distributive property.

$x = x \cdot x + x \cdot 1 - (x + 1)$

Step 2.3.1.2

Multiply $x$ by $x$ .

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

Step 2.3.1.3

Multiply $x$ by $1$ .

$x = x^{2} + x - (x + 1)$

Step 2.3.1.4

Apply the distributive property.

$x = x^{2} + x - x - 1 \cdot 1$

Step 2.3.1.5

Multiply $- 1$ by $1$ .

$x = x^{2} + x - x - 1$

Step 2.3.2

Combine the opposite terms in $x^{2} + x - x - 1$ .

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Step 2.3.2.1

Subtract $x$ from $x$ .

$x = x^{2} + 0 - 1$

Step 2.3.2.2

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

$x = x^{2} - 1$

Step 3

Solve the equation.

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Step 3.1

Subtract $x^{2}$ from both sides of the equation.

$x - x^{2} = - 1$

Step 3.2

Add $1$ to both sides of the equation.

$x - x^{2} + 1 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4

Substitute the values $a = - 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot 1)}}{2 \cdot - 1}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 1}}{2 \cdot - 1}$

Step 3.5.1.2

Multiply $- 4 \cdot - 1 \cdot 1$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot 1}}{2 \cdot - 1}$

Step 3.5.1.2.2

Multiply $4$ by $1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4}}{2 \cdot - 1}$

Step 3.5.1.3

Add $1$ and $4$ .

$x = \frac{- 1 \pm \sqrt{5}}{2 \cdot - 1}$

Step 3.5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{5}}{- 2}$

Step 3.5.3

Simplify $\frac{- 1 \pm \sqrt{5}}{- 2}$ .

$x = \frac{1 \pm \sqrt{5}}{2}$

Step 3.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Decimal Form:

$x = 1.61803398 \dots, - 0.61803398 \dots$