Solve Using the Quadratic Formula 1/2x^2-8=16

$\frac{1}{2} x^{2} - 8 = 16$

Step 1

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

Simplify the left side.

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

$\frac{x^{2}}{2} - 8 = 16$

Subtract $16$ from both sides of the equation.

$\frac{x^{2}}{2} - 8 - 16 = 0$

Subtract $16$ from $- 8$ .

$\frac{x^{2}}{2} - 24 = 0$

Step 2

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

Apply the distributive property.

$2 (\frac{x^{2}}{2}) + 2 \cdot - 24 = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 (\frac{x^{2}}{2}) + 2 \cdot - 24 = 0$

Rewrite the expression.

$x^{2} + 2 \cdot - 24 = 0$

Multiply $2$ by $- 24$ .

$x^{2} - 48 = 0$

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 $a = 1$ , $b = 0$ , and $c = - 48$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (1 \cdot - 48)}}{2 \cdot 1}$

Step 5

Simplify.

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 1 \cdot - 48}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 48$ .

Multiply $- 4$ by $1$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot - 48}}{2 \cdot 1}$

Multiply $- 4$ by $- 48$ .

$x = \frac{0 \pm \sqrt{0 + 192}}{2 \cdot 1}$

Add $0$ and $192$ .

$x = \frac{0 \pm \sqrt{192}}{2 \cdot 1}$

Rewrite $192$ as $8^{2} \cdot 3$ .

Factor $64$ out of $192$ .

$x = \frac{0 \pm \sqrt{64 (3)}}{2 \cdot 1}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{0 \pm \sqrt{8^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{0 \pm 8 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{0 \pm 8 \sqrt{3}}{2}$

Simplify $\frac{0 \pm 8 \sqrt{3}}{2}$ .

$x = \pm 4 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm 4 \sqrt{3}$

Decimal Form:

$x = 6.92820323 \dots, - 6.92820323 \dots$

Solve Using the Quadratic Formula 1/2x^2-8=16

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