Calculus Examples

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

Step 1

Subtract $2$ from both sides of the equation.

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

Step 2

Simplify ${(x - 1)}^{2} - {(x + 1)}^{2} - 2$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

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

Step 2.1.2.2

Apply the distributive property.

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

Step 2.1.2.3

Apply the distributive property.

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

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.3.1.2

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

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

Step 2.1.3.1.3

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

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

Step 2.1.3.1.4

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

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

Step 2.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.1.3.2

Subtract $x$ from $- x$ .

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

Step 2.1.4

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

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

Step 2.1.5

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

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Step 2.1.5.1

Apply the distributive property.

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

Step 2.1.5.2

Apply the distributive property.

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

Step 2.1.5.3

Apply the distributive property.

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

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $x$ by $x$ .

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

Step 2.1.6.1.2

Multiply $x$ by $1$ .

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

Step 2.1.6.1.3

Multiply $x$ by $1$ .

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

Step 2.1.6.1.4

Multiply $1$ by $1$ .

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

Step 2.1.6.2

Add $x$ and $x$ .

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

Step 2.1.7

Apply the distributive property.

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

Step 2.1.8

Simplify.

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Step 2.1.8.1

Multiply $2$ by $- 1$ .

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

Step 2.1.8.2

Multiply $- 1$ by $1$ .

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

Step 2.2

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

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Step 2.2.1

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

$- 2 x + 1 + 0 - 2 x - 1 - 2 = 0$

Step 2.2.2

Add $- 2 x + 1$ and $0$ .

$- 2 x + 1 - 2 x - 1 - 2 = 0$

Step 2.2.3

Subtract $1$ from $1$ .

$- 2 x - 2 x + 0 - 2 = 0$

Step 2.2.4

Add $- 2 x - 2 x$ and $0$ .

$- 2 x - 2 x - 2 = 0$

Step 2.3

Subtract $2 x$ from $- 2 x$ .

$- 4 x - 2 = 0$

Step 3

Factor $- 2$ out of $- 4 x - 2$ .

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Step 3.1

Factor $- 2$ out of $- 4 x$ .

$- 2 (2 x) - 2 = 0$

Step 3.2

Factor $- 2$ out of $- 2$ .

$- 2 (2 x) - 2 \cdot 1 = 0$

Step 3.3

Factor $- 2$ out of $- 2 (2 x) - 2 (1)$ .

$- 2 (2 x + 1) = 0$

Step 4

Divide each term in $- 2 (2 x + 1) = 0$ by $- 2$ and simplify.

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Step 4.1

Divide each term in $- 2 (2 x + 1) = 0$ by $- 2$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $2 x + 1$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $- 2$ .

$2 x + 1 = 0$

Step 5

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 6

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 x = - 1$ by $2$ .

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

Divide $x$ by $1$ .

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.5$