Finite Math Examples

$x^{2} - 2 x + 82 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 82$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 82)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 82}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 82$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 82}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $82$ .

$x = \frac{2 \pm \sqrt{4 - 328}}{2 \cdot 1}$

Step 3.1.3

Subtract $328$ from $4$ .

$x = \frac{2 \pm \sqrt{- 324}}{2 \cdot 1}$

Step 3.1.4

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 1}$

Step 3.1.5

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 1}$

Step 3.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{324}}{2 \cdot 1}$

Step 3.1.7

Rewrite $324$ as $18^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 1}$

Step 3.1.8

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

$x = \frac{2 \pm i \cdot 18}{2 \cdot 1}$

Step 3.1.9

Move $18$ to the left of $i$ .

$x = \frac{2 \pm 18 i}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 18 i}{2}$

Step 3.3

Simplify $\frac{2 \pm 18 i}{2}$ .

$x = 1 \pm 9 i$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 82}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 82$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 82}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $82$ .

$x = \frac{2 \pm \sqrt{4 - 328}}{2 \cdot 1}$

Step 4.1.3

Subtract $328$ from $4$ .

$x = \frac{2 \pm \sqrt{- 324}}{2 \cdot 1}$

Step 4.1.4

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 1}$

Step 4.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{324}}{2 \cdot 1}$

Step 4.1.7

Rewrite $324$ as $18^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 1}$

Step 4.1.8

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

$x = \frac{2 \pm i \cdot 18}{2 \cdot 1}$

Step 4.1.9

Move $18$ to the left of $i$ .

$x = \frac{2 \pm 18 i}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 18 i}{2}$

Step 4.3

Simplify $\frac{2 \pm 18 i}{2}$ .

$x = 1 \pm 9 i$

Step 4.4

Change the $\pm$ to $+$ .

$x = 1 + 9 i$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 82}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 82$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 82}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $82$ .

$x = \frac{2 \pm \sqrt{4 - 328}}{2 \cdot 1}$

Step 5.1.3

Subtract $328$ from $4$ .

$x = \frac{2 \pm \sqrt{- 324}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{324}}{2 \cdot 1}$

Step 5.1.7

Rewrite $324$ as $18^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 1}$

Step 5.1.8

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

$x = \frac{2 \pm i \cdot 18}{2 \cdot 1}$

Step 5.1.9

Move $18$ to the left of $i$ .

$x = \frac{2 \pm 18 i}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 18 i}{2}$

Step 5.3

Simplify $\frac{2 \pm 18 i}{2}$ .

$x = 1 \pm 9 i$

Step 5.4

Change the $\pm$ to $-$ .

$x = 1 - 9 i$

Step 6

The final answer is the combination of both solutions.

$x = 1 + 9 i, 1 - 9 i$