Finite Math Examples

$17 - k = k^{2} + 25$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $k^{2}$ from both sides of the equation.

$17 - k - k^{2} = 25$

Step 1.2

Subtract $25$ from both sides of the equation.

$17 - k - k^{2} - 25 = 0$

Step 2

Subtract $25$ from $17$ .

$- k - k^{2} - 8 = 0$

Step 3

Reorder terms.

$- k^{2} - k - 8 = 0$

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 $a = - 1$ , $b = - 1$ , and $c = - 8$ into the quadratic formula and solve for $k$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 1 \cdot - 8)}}{2 \cdot - 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 1$ to the power of $2$ .

$k = \frac{1 \pm \sqrt{1 - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot - 8$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$k = \frac{1 \pm \sqrt{1 + 4 \cdot - 8}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $- 8$ .

$k = \frac{1 \pm \sqrt{1 - 32}}{2 \cdot - 1}$

Step 6.1.3

Subtract $32$ from $1$ .

$k = \frac{1 \pm \sqrt{- 31}}{2 \cdot - 1}$

Step 6.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$k = \frac{1 \pm \sqrt{- 1 \cdot 31}}{2 \cdot - 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$k = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot - 1}$

Step 6.1.6

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

$k = \frac{1 \pm i \sqrt{31}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$k = \frac{1 \pm i \sqrt{31}}{- 2}$

Step 6.3

Move the negative in front of the fraction.

$k = - \frac{1 \pm i \sqrt{31}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 1$ to the power of $2$ .

$k = \frac{1 \pm \sqrt{1 - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 8$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$k = \frac{1 \pm \sqrt{1 + 4 \cdot - 8}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 8$ .

$k = \frac{1 \pm \sqrt{1 - 32}}{2 \cdot - 1}$

Step 7.1.3

Subtract $32$ from $1$ .

$k = \frac{1 \pm \sqrt{- 31}}{2 \cdot - 1}$

Step 7.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$k = \frac{1 \pm \sqrt{- 1 \cdot 31}}{2 \cdot - 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$k = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot - 1}$

Step 7.1.6

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

$k = \frac{1 \pm i \sqrt{31}}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$k = \frac{1 \pm i \sqrt{31}}{- 2}$

Step 7.3

Move the negative in front of the fraction.

$k = - \frac{1 \pm i \sqrt{31}}{2}$

Step 7.4

Change the $\pm$ to $+$ .

$k = - \frac{1 + i \sqrt{31}}{2}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 1$ to the power of $2$ .

$k = \frac{1 \pm \sqrt{1 - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

Step 8.1.2

Multiply $- 4 \cdot - 1 \cdot - 8$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 1$ .

$k = \frac{1 \pm \sqrt{1 + 4 \cdot - 8}}{2 \cdot - 1}$

Step 8.1.2.2

Multiply $4$ by $- 8$ .

$k = \frac{1 \pm \sqrt{1 - 32}}{2 \cdot - 1}$

Step 8.1.3

Subtract $32$ from $1$ .

$k = \frac{1 \pm \sqrt{- 31}}{2 \cdot - 1}$

Step 8.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$k = \frac{1 \pm \sqrt{- 1 \cdot 31}}{2 \cdot - 1}$

Step 8.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$k = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot - 1}$

Step 8.1.6

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

$k = \frac{1 \pm i \sqrt{31}}{2 \cdot - 1}$

Step 8.2

Multiply $2$ by $- 1$ .

$k = \frac{1 \pm i \sqrt{31}}{- 2}$

Step 8.3

Move the negative in front of the fraction.

$k = - \frac{1 \pm i \sqrt{31}}{2}$

Step 8.4

Change the $\pm$ to $-$ .

$k = - \frac{1 - i \sqrt{31}}{2}$

Step 9

The final answer is the combination of both solutions.

$k = - \frac{1 + i \sqrt{31}}{2}, - \frac{1 - i \sqrt{31}}{2}$