Algebra Examples

$0 = - (\frac{1}{2}) x^{2} + 3 x + 8$

Step 1

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

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Step 1.1

Simplify the right side.

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Step 1.1.1

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

$0 = - \frac{x^{2}}{2} + 3 x + 8$

Step 1.2

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

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Step 1.2.1

Add $\frac{x^{2}}{2}$ to both sides of the equation.

$\frac{x^{2}}{2} = 3 x + 8$

Step 1.2.2

Subtract $3 x$ from both sides of the equation.

$\frac{x^{2}}{2} - 3 x = 8$

Step 1.2.3

Subtract $8$ from both sides of the equation.

$\frac{x^{2}}{2} - 3 x - 8 = 0$

Step 2

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

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Step 2.1

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 (- 3 x) + 2 \cdot - 8 = 0$

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

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 (- 3 x) + 2 \cdot - 8 = 0$

Step 2.2.1.2

Rewrite the expression.

$x^{2} + 2 (- 3 x) + 2 \cdot - 8 = 0$

Step 2.2.2

Multiply $- 3$ by $2$ .

$x^{2} - 6 x + 2 \cdot - 8 = 0$

Step 2.2.3

Multiply $2$ by $- 8$ .

$x^{2} - 6 x - 16 = 0$

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 $a = 1$ , $b = - 6$ , and $c = - 16$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 16)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 16}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 16$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 16}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 16$ .

$x = \frac{6 \pm \sqrt{36 + 64}}{2 \cdot 1}$

Step 5.1.3

Add $36$ and $64$ .

$x = \frac{6 \pm \sqrt{100}}{2 \cdot 1}$

Step 5.1.4

Rewrite $100$ as $10^{2}$ .

$x = \frac{6 \pm \sqrt{10^{2}}}{2 \cdot 1}$

Step 5.1.5

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

$x = \frac{6 \pm 10}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 10}{2}$

Step 5.3

Simplify $\frac{6 \pm 10}{2}$ .

$x = 3 \pm 5$

Step 6

The final answer is the combination of both solutions.

$x = 8, - 2$