Solve for x -1/2x^2=-6x+20

$- \frac{1}{2} x^{2} = - 6 x + 20$

Step 1

Simplify $- \frac{1}{2} x^{2}$ .

Rewrite.

$0 + 0 - \frac{1}{2} x^{2} = - 6 x + 20$

Simplify by adding zeros.

$- \frac{1}{2} x^{2} = - 6 x + 20$

Combine $x^{2}$ and $\frac{1}{2}$ .

$- \frac{x^{2}}{2} = - 6 x + 20$

Step 2

Add $6 x$ to both sides of the equation.

$- \frac{x^{2}}{2} + 6 x = 20$

Step 3

Subtract $20$ from both sides of the equation.

$- \frac{x^{2}}{2} + 6 x - 20 = 0$

Step 4

Multiply through by the least common denominator $2$ , then simplify.

Apply the distributive property.

$2 (- \frac{x^{2}}{2}) + 2 (6 x) + 2 \cdot - 20 = 0$

Simplify.

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x^{2}}{2}$ into the numerator.

$2 (\frac{- x^{2}}{2}) + 2 (6 x) + 2 \cdot - 20 = 0$

Cancel the common factor.

$2 (\frac{- x^{2}}{2}) + 2 (6 x) + 2 \cdot - 20 = 0$

Rewrite the expression.

$- x^{2} + 2 (6 x) + 2 \cdot - 20 = 0$

Multiply $6$ by $2$ .

$- x^{2} + 12 x + 2 \cdot - 20 = 0$

Multiply $2$ by $- 20$ .

$- x^{2} + 12 x - 40 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 1$ , $b = 12$ , and $c = - 40$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 1 \cdot - 40)}}{2 \cdot - 1}$

Step 7

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 1 \cdot - 40}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 40$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 12 \pm \sqrt{144 + 4 \cdot - 40}}{2 \cdot - 1}$

Multiply $4$ by $- 40$ .

$x = \frac{- 12 \pm \sqrt{144 - 160}}{2 \cdot - 1}$

Subtract $160$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 16}}{2 \cdot - 1}$

Rewrite $- 16$ as $- 1 (16)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 16}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{16}}{2 \cdot - 1}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{16}}{2 \cdot - 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{4^{2}}}{2 \cdot - 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 12 \pm i \cdot 4}{2 \cdot - 1}$

Move $4$ to the left of $i$ .

$x = \frac{- 12 \pm 4 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 12 \pm 4 i}{- 2}$

Simplify $\frac{- 12 \pm 4 i}{- 2}$ .

$x = 6 \pm 2 i$

Step 8

The final answer is the combination of both solutions.

$x = 6 + 2 i, 6 - 2 i$

Solve for x -1/2x^2=-6x+20

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