Finite Math Examples

$- 2 z^{2} - 2 = - 28 z$

Step 1

Add $28 z$ to both sides of the equation.

$- 2 z^{2} - 2 + 28 z = 0$

Step 2

Factor $- 2$ out of $- 2 z^{2} - 2 + 28 z$ .

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Step 2.1

Move $- 2$ .

$- 2 z^{2} + 28 z - 2 = 0$

Step 2.2

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

$- 2 z^{2} + 28 z - 2 = 0$

Step 2.3

Factor $- 2$ out of $28 z$ .

$- 2 z^{2} - 2 (- 14 z) - 2 = 0$

Step 2.4

Factor $- 2$ out of $- 2$ .

$- 2 z^{2} - 2 (- 14 z) - 2 \cdot 1 = 0$

Step 2.5

Factor $- 2$ out of $- 2 (z^{2}) - 2 (- 14 z)$ .

$- 2 (z^{2} - 14 z) - 2 \cdot 1 = 0$

Step 2.6

Factor $- 2$ out of $- 2 (z^{2} - 14 z) - 2 (1)$ .

$- 2 (z^{2} - 14 z + 1) = 0$

Step 3

Divide each term in $- 2 (z^{2} - 14 z + 1) = 0$ by $- 2$ and simplify.

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Step 3.1

Divide each term in $- 2 (z^{2} - 14 z + 1) = 0$ by $- 2$ .

$\frac{- 2 (z^{2} - 14 z + 1)}{- 2} = \frac{0}{- 2}$

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.

$\frac{- 2 (z^{2} - 14 z + 1)}{- 2} = \frac{0}{- 2}$

Step 3.2.1.2

Divide $z^{2} - 14 z + 1$ by $1$ .

$z^{2} - 14 z + 1 = \frac{0}{- 2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 2$ .

$z^{2} - 14 z + 1 = 0$

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 $a = 1$ , $b = - 14$ , and $c = 1$ into the quadratic formula and solve for $z$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 14$ to the power of $2$ .

$z = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$z = \frac{14 \pm \sqrt{196 - 4 \cdot 1}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $1$ .

$z = \frac{14 \pm \sqrt{196 - 4}}{2 \cdot 1}$

Step 6.1.3

Subtract $4$ from $196$ .

$z = \frac{14 \pm \sqrt{192}}{2 \cdot 1}$

Step 6.1.4

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 6.1.4.1

Factor $64$ out of $192$ .

$z = \frac{14 \pm \sqrt{64 (3)}}{2 \cdot 1}$

Step 6.1.4.2

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

$z = \frac{14 \pm \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Step 6.1.5

Pull terms out from under the radical.

$z = \frac{14 \pm 8 \sqrt{3}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$z = \frac{14 \pm 8 \sqrt{3}}{2}$

Step 6.3

Simplify $\frac{14 \pm 8 \sqrt{3}}{2}$ .

$z = 7 \pm 4 \sqrt{3}$

Step 7

The final answer is the combination of both solutions.

$z = 7 + 4 \sqrt{3}, 7 - 4 \sqrt{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$z = 7 + 4 \sqrt{3}, 7 - 4 \sqrt{3}$

Decimal Form:

$z = 13.92820323 \dots, 0.07179676 \dots$