Finite Math Examples

$c (x) = x^{2} - 40 x + 405$

Step 1

Set $x^{2} - 40 x + 405$ equal to $0$ .

$x^{2} - 40 x + 405 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 1$ , $b = - 40$ , and $c = 405$ into the quadratic formula and solve for $x$ .

$\frac{40 \pm \sqrt{{(- 40)}^{2} - 4 \cdot (1 \cdot 405)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 40$ to the power of $2$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 1 \cdot 405}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot 405$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{40 \pm \sqrt{1600 - 4 \cdot 405}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $405$ .

$x = \frac{40 \pm \sqrt{1600 - 1620}}{2 \cdot 1}$

Step 2.3.1.3

Subtract $1620$ from $1600$ .

$x = \frac{40 \pm \sqrt{- 20}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{40 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 1}$

Step 2.3.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{40 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 1}$

Step 2.3.1.6

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

$x = \frac{40 \pm i \cdot \sqrt{20}}{2 \cdot 1}$

Step 2.3.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 2.3.1.7.1

Factor $4$ out of $20$ .

$x = \frac{40 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 1}$

Step 2.3.1.7.2

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

$x = \frac{40 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Step 2.3.1.8

Pull terms out from under the radical.

$x = \frac{40 \pm i \cdot (2 \sqrt{5})}{2 \cdot 1}$

Step 2.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{40 \pm 2 i \sqrt{5}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{40 \pm 2 i \sqrt{5}}{2}$

Step 2.3.3

Simplify $\frac{40 \pm 2 i \sqrt{5}}{2}$ .

$x = 20 \pm i \sqrt{5}$

Step 2.4

The final answer is the combination of both solutions.

$x = 20 + i \sqrt{5}, 20 - i \sqrt{5}$

$x = 20 \pm i \sqrt{5}$

Step 3