Precalculus Examples

$16 z^{2} + 62 = 64 z$

Step 1

Subtract $64 z$ from both sides of the equation.

$16 z^{2} + 62 - 64 z = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor $2$ out of $16 z^{2} + 62 - 64 z$ .

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Step 2.1.1

Factor $2$ out of $16 z^{2}$ .

$2 (8 z^{2}) + 62 - 64 z = 0$

Step 2.1.2

Factor $2$ out of $62$ .

$2 (8 z^{2}) + 2 (31) - 64 z = 0$

Step 2.1.3

Factor $2$ out of $- 64 z$ .

$2 (8 z^{2}) + 2 (31) + 2 (- 32 z) = 0$

Step 2.1.4

Factor $2$ out of $2 (8 z^{2}) + 2 (31)$ .

$2 (8 z^{2} + 31) + 2 (- 32 z) = 0$

Step 2.1.5

Factor $2$ out of $2 (8 z^{2} + 31) + 2 (- 32 z)$ .

$2 (8 z^{2} + 31 - 32 z) = 0$

Step 2.2

Reorder terms.

$2 (8 z^{2} - 32 z + 31) = 0$

Step 3

Divide each term in $2 (8 z^{2} - 32 z + 31) = 0$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 (8 z^{2} - 32 z + 31) = 0$ by $2$ .

$\frac{2 (8 z^{2} - 32 z + 31)}{2} = \frac{0}{2}$

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.

$\frac{2 (8 z^{2} - 32 z + 31)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $8 z^{2} - 32 z + 31$ by $1$ .

$8 z^{2} - 32 z + 31 = \frac{0}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $2$ .

$8 z^{2} - 32 z + 31 = 0$

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 $a = 8$ , $b = - 32$ , and $c = 31$ into the quadratic formula and solve for $z$ .

$\frac{32 \pm \sqrt{{(- 32)}^{2} - 4 \cdot (8 \cdot 31)}}{2 \cdot 8}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 32$ to the power of $2$ .

$z = \frac{32 \pm \sqrt{1024 - 4 \cdot 8 \cdot 31}}{2 \cdot 8}$

Step 6.1.2

Multiply $- 4 \cdot 8 \cdot 31$ .

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Step 6.1.2.1

Multiply $- 4$ by $8$ .

$z = \frac{32 \pm \sqrt{1024 - 32 \cdot 31}}{2 \cdot 8}$

Step 6.1.2.2

Multiply $- 32$ by $31$ .

$z = \frac{32 \pm \sqrt{1024 - 992}}{2 \cdot 8}$

Step 6.1.3

Subtract $992$ from $1024$ .

$z = \frac{32 \pm \sqrt{32}}{2 \cdot 8}$

Step 6.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 6.1.4.1

Factor $16$ out of $32$ .

$z = \frac{32 \pm \sqrt{16 (2)}}{2 \cdot 8}$

Step 6.1.4.2

Rewrite $16$ as $4^{2}$ .

$z = \frac{32 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 8}$

Step 6.1.5

Pull terms out from under the radical.

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

Step 6.2

Multiply $2$ by $8$ .

$z = \frac{32 \pm 4 \sqrt{2}}{16}$

Step 6.3

Simplify $\frac{32 \pm 4 \sqrt{2}}{16}$ .

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

Step 7

The final answer is the combination of both solutions.

$z = \frac{8 + \sqrt{2}}{4}, \frac{8 - \sqrt{2}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$z = \frac{8 + \sqrt{2}}{4}, \frac{8 - \sqrt{2}}{4}$

Decimal Form:

$z = 2.35355339 \dots, 1.64644660 \dots$