Algebra Examples

$7 - 8 z^{2} = 6 z + 16$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Move all the expressions to the left side of the equation.

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Step 1.1.1

Subtract $6 z$ from both sides of the equation.

$7 - 8 z^{2} - 6 z = 16$

Step 1.1.2

Subtract $16$ from both sides of the equation.

$7 - 8 z^{2} - 6 z - 16 = 0$

Step 1.2

Subtract $16$ from $7$ .

$- 8 z^{2} - 6 z - 9 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 8$ , $b = - 6$ , and $c = - 9$ into the quadratic formula and solve for $z$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (- 8 \cdot - 9)}}{2 \cdot - 8}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 6$ to the power of $2$ .

$z = \frac{6 \pm \sqrt{36 - 4 \cdot - 8 \cdot - 9}}{2 \cdot - 8}$

Step 4.1.2

Multiply $- 4 \cdot - 8 \cdot - 9$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 8$ .

$z = \frac{6 \pm \sqrt{36 + 32 \cdot - 9}}{2 \cdot - 8}$

Step 4.1.2.2

Multiply $32$ by $- 9$ .

$z = \frac{6 \pm \sqrt{36 - 288}}{2 \cdot - 8}$

Step 4.1.3

Subtract $288$ from $36$ .

$z = \frac{6 \pm \sqrt{- 252}}{2 \cdot - 8}$

Step 4.1.4

Rewrite $- 252$ as $- 1 (252)$ .

$z = \frac{6 \pm \sqrt{- 1 \cdot 252}}{2 \cdot - 8}$

Step 4.1.5

Rewrite $\sqrt{- 1 (252)}$ as $\sqrt{- 1} \cdot \sqrt{252}$ .

$z = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{252}}{2 \cdot - 8}$

Step 4.1.6

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

$z = \frac{6 \pm i \cdot \sqrt{252}}{2 \cdot - 8}$

Step 4.1.7

Rewrite $252$ as $6^{2} \cdot 7$ .

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Step 4.1.7.1

Factor $36$ out of $252$ .

$z = \frac{6 \pm i \cdot \sqrt{36 (7)}}{2 \cdot - 8}$

Step 4.1.7.2

Rewrite $36$ as $6^{2}$ .

$z = \frac{6 \pm i \cdot \sqrt{6^{2} \cdot 7}}{2 \cdot - 8}$

Step 4.1.8

Pull terms out from under the radical.

$z = \frac{6 \pm i \cdot (6 \sqrt{7})}{2 \cdot - 8}$

Step 4.1.9

Move $6$ to the left of $i$ .

$z = \frac{6 \pm 6 i \sqrt{7}}{2 \cdot - 8}$

Step 4.2

Multiply $2$ by $- 8$ .

$z = \frac{6 \pm 6 i \sqrt{7}}{- 16}$

Step 4.3

Simplify $\frac{6 \pm 6 i \sqrt{7}}{- 16}$ .

$z = \frac{3 \pm 3 i \sqrt{7}}{- 8}$

Step 4.4

Move the negative in front of the fraction.

$z = - \frac{3 \pm 3 i \sqrt{7}}{8}$