Basic Math Examples

m^{3} - 1 = 0

$m^{3} - 1 = 0$

Step 1

Add

1

$1$ to both sides of the equation.

m^{3} = 1

$m^{3} = 1$

Step 2

Subtract

1

$1$ from both sides of the equation.

m^{3} - 1 = 0

$m^{3} - 1 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Rewrite

1

$1$ as

1^{3}

$1^{3}$ .

m^{3} - 1^{3} = 0

$m^{3} - 1^{3} = 0$

Step 3.2

Since both terms are perfect cubes, factor using the difference of cubes formula,

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

a = m

$a = m$ and

b = 1

$b = 1$ .

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

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

Step 3.3

Simplify.

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Step 3.3.1

Multiply

m

$m$ by

1

$1$ .

(m - 1) (m^{2} + m + 1^{2}) = 0

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

Step 3.3.2

One to any power is one.

(m - 1) (m^{2} + m + 1) = 0

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

(m - 1) (m^{2} + m + 1) = 0

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

(m - 1) (m^{2} + m + 1) = 0

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

Step 4

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

m - 1 = 0

$m - 1 = 0$

m^{2} + m + 1 = 0

$m^{2} + m + 1 = 0$

Step 5

Set

m - 1

$m - 1$ equal to

0

$0$ and solve for

m

$m$ .

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Step 5.1

Set

m - 1

$m - 1$ equal to

0

$0$ .

m - 1 = 0

$m - 1 = 0$

Step 5.2

Add

1

$1$ to both sides of the equation.

m = 1

$m = 1$

m = 1

$m = 1$

Step 6

Set

m^{2} + m + 1

$m^{2} + m + 1$ equal to

0

$0$ and solve for

m

$m$ .

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Step 6.1

Set

m^{2} + m + 1

$m^{2} + m + 1$ equal to

0

$0$ .

m^{2} + m + 1 = 0

$m^{2} + m + 1 = 0$

Step 6.2

Solve

m^{2} + m + 1 = 0

$m^{2} + m + 1 = 0$ for

m

$m$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

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

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

Step 6.2.2

Substitute the values

a = 1

$a = 1$ ,

b = 1

$b = 1$ , and

c = 1

$c = 1$ into the quadratic formula and solve for

m

$m$ .

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

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

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

One to any power is one.

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

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

Step 6.2.3.1.2

Multiply

- 4 \cdot 1 \cdot 1

$- 4 \cdot 1 \cdot 1$ .

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Step 6.2.3.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

Step 6.2.3.1.2.2

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

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

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

Step 6.2.3.1.3

Subtract

4

$4$ from

1

$1$ .

m = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}

$m = \frac{- 1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 6.2.3.1.4

Rewrite

- 3

$- 3$ as

- 1 (3)

$- 1 (3)$ .

m = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}

$m = \frac{- 1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 6.2.3.1.5

Rewrite

\sqrt{- 1 (3)}

$\sqrt{- 1 (3)}$ as

\sqrt{- 1} \cdot \sqrt{3}

$\sqrt{- 1} \cdot \sqrt{3}$ .

m = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}

$m = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 6.2.3.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

m = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}

$m = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

m = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}

$m = \frac{- 1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 6.2.3.2

Multiply

2

$2$ by

1

$1$ .

m = \frac{- 1 \pm i \sqrt{3}}{2}

$m = \frac{- 1 \pm i \sqrt{3}}{2}$

m = \frac{- 1 \pm i \sqrt{3}}{2}

$m = \frac{- 1 \pm i \sqrt{3}}{2}$

Step 6.2.4

The final answer is the combination of both solutions.

m = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}

$m = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

m = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}

$m = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

m = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}

$m = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 7

The final solution is all the values that make

(m - 1) (m^{2} + m + 1) = 0

$(m - 1) (m^{2} + m + 1) = 0$ true.

m = 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}

$m = 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$