Pre-Algebra Examples

$8 z^{2} - 8 z = - 38$

Step 1

Add $38$ to both sides of the equation.

$8 z^{2} - 8 z + 38 = 0$

Step 2

Factor $2$ out of $8 z^{2} - 8 z + 38$ .

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Step 2.1

Factor $2$ out of $8 z^{2}$ .

$2 (4 z^{2}) - 8 z + 38 = 0$

Step 2.2

Factor $2$ out of $- 8 z$ .

$2 (4 z^{2}) + 2 (- 4 z) + 38 = 0$

Step 2.3

Factor $2$ out of $38$ .

$2 (4 z^{2}) + 2 (- 4 z) + 2 (19) = 0$

Step 2.4

Factor $2$ out of $2 (4 z^{2}) + 2 (- 4 z)$ .

$2 (4 z^{2} - 4 z) + 2 (19) = 0$

Step 2.5

Factor $2$ out of $2 (4 z^{2} - 4 z) + 2 (19)$ .

$2 (4 z^{2} - 4 z + 19) = 0$

Step 3

Divide each term in $2 (4 z^{2} - 4 z + 19) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (4 z^{2} - 4 z + 19) = 0$ by $2$ .

$\frac{2 (4 z^{2} - 4 z + 19)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 (4 z^{2} - 4 z + 19)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $4 z^{2} - 4 z + 19$ by $1$ .

$4 z^{2} - 4 z + 19 = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$4 z^{2} - 4 z + 19 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = - 4$ , and $c = 19$ into the quadratic formula and solve for $z$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (4 \cdot 19)}}{2 \cdot 4}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot 19}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot 19$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$z = \frac{4 \pm \sqrt{16 - 16 \cdot 19}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $19$ .

$z = \frac{4 \pm \sqrt{16 - 304}}{2 \cdot 4}$

Step 6.1.3

Subtract $304$ from $16$ .

$z = \frac{4 \pm \sqrt{- 288}}{2 \cdot 4}$

Step 6.1.4

Rewrite $- 288$ as $- 1 (288)$ .

$z = \frac{4 \pm \sqrt{- 1 \cdot 288}}{2 \cdot 4}$

Step 6.1.5

Rewrite $\sqrt{- 1 (288)}$ as $\sqrt{- 1} \cdot \sqrt{288}$ .

$z = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{288}}{2 \cdot 4}$

Step 6.1.6

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

$z = \frac{4 \pm i \cdot \sqrt{288}}{2 \cdot 4}$

Step 6.1.7

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 6.1.7.1

Factor $144$ out of $288$ .

$z = \frac{4 \pm i \cdot \sqrt{144 (2)}}{2 \cdot 4}$

Step 6.1.7.2

Rewrite $144$ as $12^{2}$ .

$z = \frac{4 \pm i \cdot \sqrt{12^{2} \cdot 2}}{2 \cdot 4}$

Step 6.1.8

Pull terms out from under the radical.

$z = \frac{4 \pm i \cdot (12 \sqrt{2})}{2 \cdot 4}$

Step 6.1.9

Move $12$ to the left of $i$ .

$z = \frac{4 \pm 12 i \sqrt{2}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$z = \frac{4 \pm 12 i \sqrt{2}}{8}$

Step 6.3

Simplify $\frac{4 \pm 12 i \sqrt{2}}{8}$ .

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

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot 19}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot 19$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

$z = \frac{4 \pm \sqrt{16 - 16 \cdot 19}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $19$ .

$z = \frac{4 \pm \sqrt{16 - 304}}{2 \cdot 4}$

Step 7.1.3

Subtract $304$ from $16$ .

$z = \frac{4 \pm \sqrt{- 288}}{2 \cdot 4}$

Step 7.1.4

Rewrite $- 288$ as $- 1 (288)$ .

$z = \frac{4 \pm \sqrt{- 1 \cdot 288}}{2 \cdot 4}$

Step 7.1.5

Rewrite $\sqrt{- 1 (288)}$ as $\sqrt{- 1} \cdot \sqrt{288}$ .

$z = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{288}}{2 \cdot 4}$

Step 7.1.6

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

$z = \frac{4 \pm i \cdot \sqrt{288}}{2 \cdot 4}$

Step 7.1.7

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 7.1.7.1

Factor $144$ out of $288$ .

$z = \frac{4 \pm i \cdot \sqrt{144 (2)}}{2 \cdot 4}$

Step 7.1.7.2

Rewrite $144$ as $12^{2}$ .

$z = \frac{4 \pm i \cdot \sqrt{12^{2} \cdot 2}}{2 \cdot 4}$

Step 7.1.8

Pull terms out from under the radical.

$z = \frac{4 \pm i \cdot (12 \sqrt{2})}{2 \cdot 4}$

Step 7.1.9

Move $12$ to the left of $i$ .

$z = \frac{4 \pm 12 i \sqrt{2}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$z = \frac{4 \pm 12 i \sqrt{2}}{8}$

Step 7.3

Simplify $\frac{4 \pm 12 i \sqrt{2}}{8}$ .

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

Step 7.4

Change the $\pm$ to $+$ .

$z = \frac{1 + 3 i \sqrt{2}}{2}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

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

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 4 \cdot 19}}{2 \cdot 4}$

Step 8.1.2

Multiply $- 4 \cdot 4 \cdot 19$ .

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Step 8.1.2.1

Multiply $- 4$ by $4$ .

$z = \frac{4 \pm \sqrt{16 - 16 \cdot 19}}{2 \cdot 4}$

Step 8.1.2.2

Multiply $- 16$ by $19$ .

$z = \frac{4 \pm \sqrt{16 - 304}}{2 \cdot 4}$

Step 8.1.3

Subtract $304$ from $16$ .

$z = \frac{4 \pm \sqrt{- 288}}{2 \cdot 4}$

Step 8.1.4

Rewrite $- 288$ as $- 1 (288)$ .

$z = \frac{4 \pm \sqrt{- 1 \cdot 288}}{2 \cdot 4}$

Step 8.1.5

Rewrite $\sqrt{- 1 (288)}$ as $\sqrt{- 1} \cdot \sqrt{288}$ .

$z = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{288}}{2 \cdot 4}$

Step 8.1.6

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

$z = \frac{4 \pm i \cdot \sqrt{288}}{2 \cdot 4}$

Step 8.1.7

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 8.1.7.1

Factor $144$ out of $288$ .

$z = \frac{4 \pm i \cdot \sqrt{144 (2)}}{2 \cdot 4}$

Step 8.1.7.2

Rewrite $144$ as $12^{2}$ .

$z = \frac{4 \pm i \cdot \sqrt{12^{2} \cdot 2}}{2 \cdot 4}$

Step 8.1.8

Pull terms out from under the radical.

$z = \frac{4 \pm i \cdot (12 \sqrt{2})}{2 \cdot 4}$

Step 8.1.9

Move $12$ to the left of $i$ .

$z = \frac{4 \pm 12 i \sqrt{2}}{2 \cdot 4}$

Step 8.2

Multiply $2$ by $4$ .

$z = \frac{4 \pm 12 i \sqrt{2}}{8}$

Step 8.3

Simplify $\frac{4 \pm 12 i \sqrt{2}}{8}$ .

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

Step 8.4

Change the $\pm$ to $-$ .

$z = \frac{1 - 3 i \sqrt{2}}{2}$

Step 9

The final answer is the combination of both solutions.

$z = \frac{1 + 3 i \sqrt{2}}{2}, \frac{1 - 3 i \sqrt{2}}{2}$