Algebra Examples

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

Step 1

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

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Step 1.1

Simplify the left side.

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Step 1.1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

Subtract $20$ from both sides of the equation.

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factor.

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

Step 2.2.1.3

Rewrite the expression.

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

Step 2.2.2

Multiply $6$ by $2$ .

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

Step 2.2.3

Multiply $2$ by $- 20$ .

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

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

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 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 5.1.2.2

Multiply $4$ by $- 40$ .

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

Step 5.1.3

Subtract $160$ from $144$ .

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.1.7

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

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

Step 5.1.8

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

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

Step 5.1.9

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

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

Step 5.2

Multiply $2$ by $- 1$ .

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

Step 5.3

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

$x = 6 \pm 2 i$