Algebra Examples

$f (x) = - \frac{1}{2} {(x - 1)}^{2} + 2$

Step 1

Set $- \frac{1}{2} \cdot {(x - 1)}^{2} + 2$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Subtract $2$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

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

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

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 $- 2 (- \frac{{(x - 1)}^{2}}{2})$ .

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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 $- \frac{{(x - 1)}^{2}}{2}$ into the numerator.

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

Step 2.4.1.1.1.2

Factor $2$ out of $- 2$ .

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

Step 2.4.1.1.1.3

Cancel the common factor.

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

Step 2.4.1.1.1.4

Rewrite the expression.

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

Step 2.4.1.1.2

Multiply.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.4.1.1.2.2

Multiply ${(x - 1)}^{2}$ by $1$ .

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $- 2$ by $- 2$ .

${(x - 1)}^{2} = 4$

Step 2.5

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

$x - 1 = \pm \sqrt{4}$

Step 2.6

Simplify $\pm \sqrt{4}$ .

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Step 2.6.1

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

$x - 1 = \pm \sqrt{2^{2}}$

Step 2.6.2

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

$x - 1 = \pm 2$

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.

$x - 1 = 2$

Step 2.7.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.7.2.1

Add $1$ to both sides of the equation.

$x = 2 + 1$

Step 2.7.2.2

Add $2$ and $1$ .

$x = 3$

Step 2.7.3

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

$x - 1 = - 2$

Step 2.7.4

Move all terms not containing $x$ to the right side of the equation.

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Step 2.7.4.1

Add $1$ to both sides of the equation.

$x = - 2 + 1$

Step 2.7.4.2

Add $- 2$ and $1$ .

$x = - 1$

Step 2.7.5

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

$x = 3, - 1$

Step 3