Linear Algebra Examples

$- 5 k^{2} - k + 5 = 9$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $9$ from both sides of the equation.

$- 5 k^{2} - k + 5 - 9 = 0$

Step 1.2

Subtract $9$ from $5$ .

$- 5 k^{2} - k - 4 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 5$ , $b = - 1$ , and $c = - 4$ into the quadratic formula and solve for $k$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 5 \cdot - 4)}}{2 \cdot - 5}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$k = \frac{1 \pm \sqrt{1 - 4 \cdot - 5 \cdot - 4}}{2 \cdot - 5}$

Step 4.1.2

Multiply $- 4 \cdot - 5 \cdot - 4$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 5$ .

$k = \frac{1 \pm \sqrt{1 + 20 \cdot - 4}}{2 \cdot - 5}$

Step 4.1.2.2

Multiply $20$ by $- 4$ .

$k = \frac{1 \pm \sqrt{1 - 80}}{2 \cdot - 5}$

Step 4.1.3

Subtract $80$ from $1$ .

$k = \frac{1 \pm \sqrt{- 79}}{2 \cdot - 5}$

Step 4.1.4

Rewrite $- 79$ as $- 1 (79)$ .

$k = \frac{1 \pm \sqrt{- 1 \cdot 79}}{2 \cdot - 5}$

Step 4.1.5

Rewrite $\sqrt{- 1 (79)}$ as $\sqrt{- 1} \cdot \sqrt{79}$ .

$k = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{79}}{2 \cdot - 5}$

Step 4.1.6

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

$k = \frac{1 \pm i \sqrt{79}}{2 \cdot - 5}$

Step 4.2

Multiply $2$ by $- 5$ .

$k = \frac{1 \pm i \sqrt{79}}{- 10}$

Step 4.3

Move the negative in front of the fraction.

$k = - \frac{1 \pm i \sqrt{79}}{10}$

Step 5