Precalculus Examples

$f (x) = x^{3} - 2 x^{2} + 2 x$

Step 1

Set $x^{3} - 2 x^{2} + 2 x$ equal to $0$ .

$x^{3} - 2 x^{2} + 2 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x$ out of $x^{3} - 2 x^{2} + 2 x$ .

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Step 2.1.1

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

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

Step 2.1.2

Factor $x$ out of $- 2 x^{2}$ .

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

Step 2.1.3

Factor $x$ out of $2 x$ .

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

Step 2.1.4

Factor $x$ out of $x \cdot x^{2} + x (- 2 x)$ .

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

Step 2.1.5

Factor $x$ out of $x (x^{2} - 2 x) + x \cdot 2$ .

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

Step 2.2

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

$x = 0$

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

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x^{2} - 2 x + 2$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - 2 x + 2$ equal to $0$ .

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

Step 2.4.2

Solve $x^{2} - 2 x + 2 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

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

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

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Subtract $8$ from $4$ .

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

Step 2.4.2.3.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.4.2.3.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.4.2.3.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.4.2.3.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 2.4.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 2.4.2.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 2.4.2.3.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 2.4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.2.4.1

Simplify the numerator.

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Step 2.4.2.4.1.1

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

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

Step 2.4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 2.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.4.2.4.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 2.4.2.4.1.3

Subtract $8$ from $4$ .

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

Step 2.4.2.4.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.4.2.4.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.4.2.4.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.4.2.4.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 2.4.2.4.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 2.4.2.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 2.4.2.4.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 2.4.2.4.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 2.4.2.4.4

Change the $\pm$ to $+$ .

$x = 1 + i$

Step 2.4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.4.2.5.1

Simplify the numerator.

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Step 2.4.2.5.1.1

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

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

Step 2.4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 2.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.4.2.5.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 2.4.2.5.1.3

Subtract $8$ from $4$ .

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

Step 2.4.2.5.1.4

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 2.4.2.5.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

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

Step 2.4.2.5.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 2.4.2.5.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 2.4.2.5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{2 \pm i \cdot 2}{2 \cdot 1}$

Step 2.4.2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{2 \pm 2 i}{2 \cdot 1}$

Step 2.4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 2 i}{2}$

Step 2.4.2.5.3

Simplify $\frac{2 \pm 2 i}{2}$ .

$x = 1 \pm i$

Step 2.4.2.5.4

Change the $\pm$ to $-$ .

$x = 1 - i$

Step 2.4.2.6

The final answer is the combination of both solutions.

$x = 1 + i, 1 - i$

Step 2.5

The final solution is all the values that make $x (x^{2} - 2 x + 2) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $1$ )

$x = 1 + i$ (Multiplicity of $1$ )

$x = 1 - i$ (Multiplicity of $1$ )

$x = 0$ (Multiplicity of $1$ )

$x = 1 + i$ (Multiplicity of $1$ )

$x = 1 - i$ (Multiplicity of $1$ )

Step 3