Precalculus Examples

$f (x) = 13 x - 185 x^{2} + 608 x - 580$

Step 1

Set $13 x - 185 x^{2} + 608 x - 580$ equal to $0$ .

$13 x - 185 x^{2} + 608 x - 580 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $13 x$ and $608 x$ .

$- 185 x^{2} + 621 x - 580 = 0$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = - 185$ , $b = 621$ , and $c = - 580$ into the quadratic formula and solve for $x$ .

$\frac{- 621 \pm \sqrt{621^{2} - 4 \cdot (- 185 \cdot - 580)}}{2 \cdot - 185}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $621$ to the power of $2$ .

$x = \frac{- 621 \pm \sqrt{385641 - 4 \cdot - 185 \cdot - 580}}{2 \cdot - 185}$

Step 2.4.1.2

Multiply $- 4 \cdot - 185 \cdot - 580$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $- 185$ .

$x = \frac{- 621 \pm \sqrt{385641 + 740 \cdot - 580}}{2 \cdot - 185}$

Step 2.4.1.2.2

Multiply $740$ by $- 580$ .

$x = \frac{- 621 \pm \sqrt{385641 - 429200}}{2 \cdot - 185}$

Step 2.4.1.3

Subtract $429200$ from $385641$ .

$x = \frac{- 621 \pm \sqrt{- 43559}}{2 \cdot - 185}$

Step 2.4.1.4

Rewrite $- 43559$ as $- 1 (43559)$ .

$x = \frac{- 621 \pm \sqrt{- 1 \cdot 43559}}{2 \cdot - 185}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (43559)}$ as $\sqrt{- 1} \cdot \sqrt{43559}$ .

$x = \frac{- 621 \pm \sqrt{- 1} \cdot \sqrt{43559}}{2 \cdot - 185}$

Step 2.4.1.6

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

$x = \frac{- 621 \pm i \sqrt{43559}}{2 \cdot - 185}$

Step 2.4.2

Multiply $2$ by $- 185$ .

$x = \frac{- 621 \pm i \sqrt{43559}}{- 370}$

Step 2.4.3

Simplify $\frac{- 621 \pm i \sqrt{43559}}{- 370}$ .

$x = \frac{621 \pm i \sqrt{43559}}{370}$

Step 2.5

The final answer is the combination of both solutions.

$x = \frac{621 + i \sqrt{43559}}{370}, \frac{621 - i \sqrt{43559}}{370}$

$x = \frac{621 \pm i \sqrt{43559}}{370}$

Step 3