Algebra Examples

$r (x) = - \frac{1}{2} x^{2} - 24$

Step 1

Set $- \frac{1}{2} x^{2} - 24$ equal to $0$ .

$- \frac{1}{2} x^{2} - 24 = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

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

Step 2.2

Add $24$ to both sides of the equation.

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

Step 2.3

Multiply both sides of the equation by $- 2$ .

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

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Simplify $- 2 (- \frac{x^{2}}{2})$ .

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Step 2.4.1.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1.1

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

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

Step 2.4.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- x^{2}}{2} = - 2 \cdot 24$

Step 2.4.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- x^{2}}{2} = - 2 \cdot 24$

Step 2.4.1.1.1.4

Rewrite the expression.

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

Step 2.4.1.1.2

Multiply.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 2 \cdot 24$

Step 2.4.1.1.2.2

Multiply $x^{2}$ by $1$ .

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $- 2$ by $24$ .

$x^{2} = - 48$

Step 2.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 48}$

Step 2.6

Simplify $\pm \sqrt{- 48}$ .

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Step 2.6.1

Rewrite $- 48$ as $- 1 (48)$ .

$x = \pm \sqrt{- 1 (48)}$

Step 2.6.2

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{48}$

Step 2.6.3

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

$x = \pm i \cdot \sqrt{48}$

Step 2.6.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.6.4.1

Factor $16$ out of $48$ .

$x = \pm i \cdot \sqrt{16 (3)}$

Step 2.6.4.2

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

$x = \pm i \cdot \sqrt{4^{2} \cdot 3}$

Step 2.6.5

Pull terms out from under the radical.

$x = \pm i \cdot (4 \sqrt{3})$

Step 2.6.6

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

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

Step 2.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 4 i \sqrt{3}$

Step 2.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 4 i \sqrt{3}$

Step 2.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4 i \sqrt{3}, - 4 i \sqrt{3}$

Step 3