Algebra Examples

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

Step 1

Factor each term.

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Step 1.1

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

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

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 $a = x$ and $b = 1$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$(x + 1) (x - 1), 1$

Step 2.2

The LCM of one and any expression is the expression.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $(x + 1) (x - 1)$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $(x + 1) (x - 1)$ by $1$ .

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

Step 3.3.2

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

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Step 3.3.2.1

Apply the distributive property.

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

Step 3.3.2.2

Apply the distributive property.

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

Step 3.3.2.3

Apply the distributive property.

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

Step 3.3.3

Simplify and combine like terms.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply $x$ by $x$ .

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

Step 3.3.3.1.2

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

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

Step 3.3.3.1.3

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

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

Step 3.3.3.1.4

Multiply $x$ by $1$ .

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

Step 3.3.3.1.5

Multiply $- 1$ by $1$ .

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

Step 3.3.3.2

Add $- x$ and $x$ .

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

Step 3.3.3.3

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

$1 = x^{2} - 1$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $x^{2} - 1 = 1$ .

$x^{2} - 1 = 1$

Step 4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 4.2.1

Add $1$ to both sides of the equation.

$x^{2} = 1 + 1$

Step 4.2.2

Add $1$ and $1$ .

$x^{2} = 2$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

Step 4.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 4.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 4.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 1.41421356 \dots, - 1.41421356 \dots$