Basic Math Examples

z^{2} - z - 1 = 0

$z^{2} - z - 1 = 0$

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

z

$z$ .

\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

$2$ .

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

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

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

$1$ .

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

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

Step 3.1.2.2

Multiply

- 4

$- 4$ by

- 1

$- 1$ .

z = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}

$z = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

z = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}

$z = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot 1}$

Step 3.1.3

Add

1

$1$ and

4

$4$ .

z = \frac{1 \pm \sqrt{5}}{2 \cdot 1}

$z = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

z = \frac{1 \pm \sqrt{5}}{2 \cdot 1}

$z = \frac{1 \pm \sqrt{5}}{2 \cdot 1}$

Step 3.2

Multiply

2

$2$ by

1

$1$ .

z = \frac{1 \pm \sqrt{5}}{2}

$z = \frac{1 \pm \sqrt{5}}{2}$

z = \frac{1 \pm \sqrt{5}}{2}

$z = \frac{1 \pm \sqrt{5}}{2}$

Step 4

The final answer is the combination of both solutions.

z = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}

$z = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

z = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}

$z = \frac{1 + \sqrt{5}}{2}, \frac{1 - \sqrt{5}}{2}$

Decimal Form:

z = 1.61803398 \dots, - 0.61803398 \dots

$z = 1.61803398 \dots, - 0.61803398 \dots$