Algebra Examples

${(x + 1)}^{3} - {(x - 1)}^{3} = 6 x (x - 3)$

Step 1

Simplify $6 x (x - 3)$ .

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Step 1.1

Apply the distributive property.

${(x + 1)}^{3} - {(x - 1)}^{3} = 6 x \cdot x + 6 x \cdot - 3$

Step 1.2

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

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Step 1.2.1

Move $x$ .

${(x + 1)}^{3} - {(x - 1)}^{3} = 6 (x \cdot x) + 6 x \cdot - 3$

Step 1.2.2

Multiply $x$ by $x$ .

${(x + 1)}^{3} - {(x - 1)}^{3} = 6 x^{2} + 6 x \cdot - 3$

Step 1.3

Multiply $- 3$ by $6$ .

${(x + 1)}^{3} - {(x - 1)}^{3} = 6 x^{2} - 18 x$

Step 2

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

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Step 2.1

Subtract $6 x^{2}$ from both sides of the equation.

${(x + 1)}^{3} - {(x - 1)}^{3} - 6 x^{2} = - 18 x$

Step 2.2

Add $18 x$ to both sides of the equation.

${(x + 1)}^{3} - {(x - 1)}^{3} - 6 x^{2} + 18 x = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Use the Binomial Theorem.

$x^{3} + 3 x^{2} \cdot 1 + 3 x \cdot 1^{2} + 1^{3} - {(x - 1)}^{3} - 6 x^{2} + 18 x = 0$

Step 2.3.2

Simplify each term.

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Step 2.3.2.1

Multiply $3$ by $1$ .

$x^{3} + 3 x^{2} + 3 x \cdot 1^{2} + 1^{3} - {(x - 1)}^{3} - 6 x^{2} + 18 x = 0$

Step 2.3.2.2

One to any power is one.

$x^{3} + 3 x^{2} + 3 x \cdot 1 + 1^{3} - {(x - 1)}^{3} - 6 x^{2} + 18 x = 0$

Step 2.3.2.3

Multiply $3$ by $1$ .

$x^{3} + 3 x^{2} + 3 x + 1^{3} - {(x - 1)}^{3} - 6 x^{2} + 18 x = 0$

Step 2.3.2.4

One to any power is one.

$x^{3} + 3 x^{2} + 3 x + 1 - {(x - 1)}^{3} - 6 x^{2} + 18 x = 0$

Step 2.3.3

Use the Binomial Theorem.

$x^{3} + 3 x^{2} + 3 x + 1 - (x^{3} + 3 x^{2} \cdot - 1 + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 6 x^{2} + 18 x = 0$

Step 2.3.4

Simplify each term.

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Step 2.3.4.1

Multiply $- 1$ by $3$ .

$x^{3} + 3 x^{2} + 3 x + 1 - (x^{3} - 3 x^{2} + 3 x {(- 1)}^{2} + {(- 1)}^{3}) - 6 x^{2} + 18 x = 0$

Step 2.3.4.2

Raise $- 1$ to the power of $2$ .

$x^{3} + 3 x^{2} + 3 x + 1 - (x^{3} - 3 x^{2} + 3 x \cdot 1 + {(- 1)}^{3}) - 6 x^{2} + 18 x = 0$

Step 2.3.4.3

Multiply $3$ by $1$ .

$x^{3} + 3 x^{2} + 3 x + 1 - (x^{3} - 3 x^{2} + 3 x + {(- 1)}^{3}) - 6 x^{2} + 18 x = 0$

Step 2.3.4.4

Raise $- 1$ to the power of $3$ .

$x^{3} + 3 x^{2} + 3 x + 1 - (x^{3} - 3 x^{2} + 3 x - 1) - 6 x^{2} + 18 x = 0$

Step 2.3.5

Apply the distributive property.

$x^{3} + 3 x^{2} + 3 x + 1 - x^{3} - (- 3 x^{2}) - (3 x) - - 1 - 6 x^{2} + 18 x = 0$

Step 2.3.6

Simplify.

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Step 2.3.6.1

Multiply $- 3$ by $- 1$ .

$x^{3} + 3 x^{2} + 3 x + 1 - x^{3} + 3 x^{2} - (3 x) - - 1 - 6 x^{2} + 18 x = 0$

Step 2.3.6.2

Multiply $3$ by $- 1$ .

$x^{3} + 3 x^{2} + 3 x + 1 - x^{3} + 3 x^{2} - 3 x - - 1 - 6 x^{2} + 18 x = 0$

Step 2.3.6.3

Multiply $- 1$ by $- 1$ .

$x^{3} + 3 x^{2} + 3 x + 1 - x^{3} + 3 x^{2} - 3 x + 1 - 6 x^{2} + 18 x = 0$

Step 2.4

Combine the opposite terms in $x^{3} + 3 x^{2} + 3 x + 1 - x^{3} + 3 x^{2} - 3 x + 1 - 6 x^{2} + 18 x$ .

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Step 2.4.1

Subtract $x^{3}$ from $x^{3}$ .

$3 x^{2} + 3 x + 1 + 0 + 3 x^{2} - 3 x + 1 - 6 x^{2} + 18 x = 0$

Step 2.4.2

Add $3 x^{2} + 3 x + 1$ and $0$ .

$3 x^{2} + 3 x + 1 + 3 x^{2} - 3 x + 1 - 6 x^{2} + 18 x = 0$

Step 2.4.3

Subtract $3 x$ from $3 x$ .

$3 x^{2} + 1 + 3 x^{2} + 0 + 1 - 6 x^{2} + 18 x = 0$

Step 2.4.4

Add $3 x^{2} + 1 + 3 x^{2}$ and $0$ .

$3 x^{2} + 1 + 3 x^{2} + 1 - 6 x^{2} + 18 x = 0$

Step 2.5

Add $3 x^{2}$ and $3 x^{2}$ .

$6 x^{2} + 1 + 1 - 6 x^{2} + 18 x = 0$

Step 2.6

Combine the opposite terms in $6 x^{2} + 1 + 1 - 6 x^{2} + 18 x$ .

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Step 2.6.1

Subtract $6 x^{2}$ from $6 x^{2}$ .

$0 + 1 + 1 + 18 x = 0$

Step 2.6.2

Add $0$ and $1$ .

$1 + 1 + 18 x = 0$

Step 2.7

Add $1$ and $1$ .

$2 + 18 x = 0$

Step 3

Subtract $2$ from both sides of the equation.

$18 x = - 2$

Step 4

Divide each term in $18 x = - 2$ by $18$ and simplify.

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Step 4.1

Divide each term in $18 x = - 2$ by $18$ .

$\frac{18 x}{18} = \frac{- 2}{18}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $18$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{18 x}{18} = \frac{- 2}{18}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{18}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 2$ and $18$ .

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Step 4.3.1.1

Factor $2$ out of $- 2$ .

$x = \frac{2 (- 1)}{18}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $18$ .

$x = \frac{2 \cdot - 1}{2 \cdot 9}$

Step 4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 1}{2 \cdot 9}$

Step 4.3.1.2.3

Rewrite the expression.

$x = \frac{- 1}{9}$

Step 4.3.2

Move the negative in front of the fraction.

$x = - \frac{1}{9}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{9}$

Decimal Form:

$x = - 0.\overline{1}$