Algebra Examples

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

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Simplify $(x - 1) (x + 1)$ .

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Step 2.1

Rewrite.

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

Step 2.2

Simplify by adding zeros.

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

Step 2.3

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

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Step 2.3.1

Apply the distributive property.

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

Step 2.3.2

Apply the distributive property.

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

Step 2.3.3

Apply the distributive property.

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

Step 2.4

Simplify terms.

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Step 2.4.1

Combine the opposite terms in $x \cdot x + x \cdot 1 - 1 x - 1 \cdot 1$ .

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Step 2.4.1.1

Reorder the factors in the terms $x \cdot 1$ and $- 1 x$ .

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

Step 2.4.1.2

Subtract $1 x$ from $1 x$ .

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

Step 2.4.1.3

Add $x \cdot x$ and $0$ .

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

Step 2.4.2

Simplify each term.

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Step 2.4.2.1

Multiply $x$ by $x$ .

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

Step 2.4.2.2

Multiply $- 1$ by $1$ .

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

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

$0 - 1 = - 1$

Step 3.2.2

Subtract $1$ from $0$ .

$- 1 = - 1$

Step 4

Since $- 1 = - 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 5

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$