Basic Math Examples

$z^{2} = 8 z - 32$

Step 1

Subtract $8 z$ from both sides of the equation.

$z^{2} - 8 z = - 32$

Step 2

Add $32$ to both sides of the equation.

$z^{2} - 8 z + 32 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 8$ , and $c = 32$ into the quadratic formula and solve for $z$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 32)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$z = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 32}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 32$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$z = \frac{8 \pm \sqrt{64 - 4 \cdot 32}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $32$ .

$z = \frac{8 \pm \sqrt{64 - 128}}{2 \cdot 1}$

Step 5.1.3

Subtract $128$ from $64$ .

$z = \frac{8 \pm \sqrt{- 64}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$z = \frac{8 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$z = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 1}$

Step 5.1.6

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

$z = \frac{8 \pm i \cdot \sqrt{64}}{2 \cdot 1}$

Step 5.1.7

Rewrite $64$ as $8^{2}$ .

$z = \frac{8 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 1}$

Step 5.1.8

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

$z = \frac{8 \pm i \cdot 8}{2 \cdot 1}$

Step 5.1.9

Move $8$ to the left of $i$ .

$z = \frac{8 \pm 8 i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$z = \frac{8 \pm 8 i}{2}$

Step 5.3

Simplify $\frac{8 \pm 8 i}{2}$ .

$z = 4 \pm 4 i$

Step 6

The final answer is the combination of both solutions.

$z = 4 + 4 i, 4 - 4 i$