Algebra Examples

x^{2} - x + 1 = 0

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

Step 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 2

Substitute the values

a = 1

$a = 1$ ,

b = - 1

$b = - 1$ , and

c = 1

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

x

$x$ .

\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 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 3.1.2

Multiply

- 4 \cdot 1 \cdot 1

$- 4 \cdot 1 \cdot 1$ .

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Step 3.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

Step 3.1.2.2

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

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

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

Step 3.1.3

Subtract

4

$4$ from

1

$1$ .

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

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

Step 3.1.4

Rewrite

- 3

$- 3$ as

- 1 (3)

$- 1 (3)$ .

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

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

Step 3.1.5

Rewrite

\sqrt{- 1 (3)}

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

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

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

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

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

Step 3.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

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

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

Step 3.2

Multiply

2

$2$ by

1

$1$ .

x = \frac{1 \pm i \sqrt{3}}{2}

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

x = \frac{1 \pm i \sqrt{3}}{2}

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