Algebra Examples

$x^{3} - 4 x^{2} + 5 x = 0$

Step 1

Factor $x$ out of $x^{3} - 4 x^{2} + 5 x$ .

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Step 1.1

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

$x \cdot x^{2} - 4 x^{2} + 5 x = 0$

Step 1.2

Factor $x$ out of $- 4 x^{2}$ .

$x \cdot x^{2} + x (- 4 x) + 5 x = 0$

Step 1.3

Factor $x$ out of $5 x$ .

$x \cdot x^{2} + x (- 4 x) + x \cdot 5 = 0$

Step 1.4

Factor $x$ out of $x \cdot x^{2} + x (- 4 x)$ .

$x (x^{2} - 4 x) + x \cdot 5 = 0$

Step 1.5

Factor $x$ out of $x (x^{2} - 4 x) + x \cdot 5$ .

$x (x^{2} - 4 x + 5) = 0$

Step 2

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

$x = 0$

$x^{2} - 4 x + 5 = 0$

Step 3

Set $x$ equal to $0$ .

$x = 0$

Step 4

Set $x^{2} - 4 x + 5$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} - 4 x + 5$ equal to $0$ .

$x^{2} - 4 x + 5 = 0$

Step 4.2

Solve $x^{2} - 4 x + 5 = 0$ for $x$ .

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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 $a = 1$ , $b = - 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

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

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 5}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $20$ from $16$ .

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

Step 4.2.3.1.4

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

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

Step 4.2.3.1.5

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

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

Step 4.2.3.1.6

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

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

Step 4.2.3.1.7

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

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

Step 4.2.3.1.8

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

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

Step 4.2.3.1.9

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

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

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i}{2}$

Step 4.2.3.3

Simplify $\frac{4 \pm 2 i}{2}$ .

$x = 2 \pm i$

Step 4.2.4

The final answer is the combination of both solutions.

$x = 2 + i, 2 - i$

Step 5

The final solution is all the values that make $x (x^{2} - 4 x + 5) = 0$ true.

$x = 0, 2 + i, 2 - i$

Step 6