Finite Math Examples

$- x^{2} + 6 x - 12 = 15$

Step 1

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

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Step 1.1

Subtract $15$ from both sides of the equation.

$- x^{2} + 6 x - 12 - 15 = 0$

Step 1.2

Subtract $15$ from $- 12$ .

$- x^{2} + 6 x - 27 = 0$

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 $a = - 1$ , $b = 6$ , and $c = - 27$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 1 \cdot - 27)}}{2 \cdot - 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 27}}{2 \cdot - 1}$

Step 4.1.2

Multiply $- 4 \cdot - 1 \cdot - 27$ .

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Step 4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 27}}{2 \cdot - 1}$

Step 4.1.2.2

Multiply $4$ by $- 27$ .

$x = \frac{- 6 \pm \sqrt{36 - 108}}{2 \cdot - 1}$

Step 4.1.3

Subtract $108$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 72}}{2 \cdot - 1}$

Step 4.1.4

Rewrite $- 72$ as $- 1 (72)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 72}}{2 \cdot - 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (72)}$ as $\sqrt{- 1} \cdot \sqrt{72}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{72}}{2 \cdot - 1}$

Step 4.1.6

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

$x = \frac{- 6 \pm i \cdot \sqrt{72}}{2 \cdot - 1}$

Step 4.1.7

Rewrite $72$ as $6^{2} \cdot 2$ .

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Step 4.1.7.1

Factor $36$ out of $72$ .

$x = \frac{- 6 \pm i \cdot \sqrt{36 (2)}}{2 \cdot - 1}$

Step 4.1.7.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{6^{2} \cdot 2}}{2 \cdot - 1}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{- 6 \pm i \cdot (6 \sqrt{2})}{2 \cdot - 1}$

Step 4.1.9

Move $6$ to the left of $i$ .

$x = \frac{- 6 \pm 6 i \sqrt{2}}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$x = \frac{- 6 \pm 6 i \sqrt{2}}{- 2}$

Step 4.3

Simplify $\frac{- 6 \pm 6 i \sqrt{2}}{- 2}$ .

$x = 3 \pm 3 i \sqrt{2}$

Step 5

The final answer is the combination of both solutions.

$x = 3 + 3 i \sqrt{2}, 3 - 3 i \sqrt{2}$