Basic Math Examples

$b^{3} + b^{2} + b = 0$

Step 1

Factor $b$ out of $b^{3} + b^{2} + b$ .

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Step 1.1

Factor $b$ out of $b^{3}$ .

$b \cdot b^{2} + b^{2} + b = 0$

Step 1.2

Factor $b$ out of $b^{2}$ .

$b \cdot b^{2} + b \cdot b + b = 0$

Step 1.3

Raise $b$ to the power of $1$ .

$b \cdot b^{2} + b \cdot b + b = 0$

Step 1.4

Factor $b$ out of $b^{1}$ .

$b \cdot b^{2} + b \cdot b + b \cdot 1 = 0$

Step 1.5

Factor $b$ out of $b \cdot b^{2} + b \cdot b$ .

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

Step 1.6

Factor $b$ out of $b (b^{2} + b) + b \cdot 1$ .

$b (b^{2} + b + 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$b = 0$

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

Step 3

Set $b$ equal to $0$ .

$b = 0$

Step 4

Set $b^{2} + b + 1$ equal to $0$ and solve for $b$ .

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Step 4.1

Set $b^{2} + b + 1$ equal to $0$ .

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

Step 4.2

Solve $b^{2} + b + 1 = 0$ for $b$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

Substitute the values $a = 1$ , $b = 1$ , and $c = 1$ into the quadratic formula and solve for $b$ .

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

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

One to any power is one.

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

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 4.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 4.2.3.1.3

Subtract $4$ from $1$ .

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

Step 4.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 4.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 4.2.3.2

Multiply $2$ by $1$ .

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

Step 4.2.4

The final answer is the combination of both solutions.

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

Step 5

The final solution is all the values that make $b (b^{2} + b + 1) = 0$ true.

$b = 0, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$