Algebra Examples

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

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Subtract $\frac{1}{2}$ from both sides of the equation.

$- \frac{1}{2} x^{2} - x = y - \frac{1}{2}$

Step 3

Complete the square.

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Step 3.1

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

$- \frac{x^{2}}{2} - x = y - \frac{1}{2}$

Step 3.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - \frac{1}{2}$

$b = - 1$

$c = 0 = y - \frac{1}{2}$

Step 3.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - \frac{1}{2}$

Step 3.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 1}{2 (- \frac{1}{2})}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Dividing two negative values results in a positive value.

$d = \frac{1}{2 (\frac{1}{2})}$

Step 3.4.2.2

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

$d = \frac{1}{\frac{2}{2}}$

Step 3.4.2.3

Divide $2$ by $2$ .

$d = \frac{1}{1}$

Step 3.4.2.4

Divide $1$ by $1$ .

$d = 1$

Step 3.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 1)}^{2}}{4 (- \frac{1}{2})}$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1}{4 (- \frac{1}{2})}$

Step 3.5.2.1.2

Simplify the denominator.

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Step 3.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - \frac{1}{- 4 (\frac{1}{2})}$

Step 3.5.2.1.2.2

Combine $- 4$ and $\frac{1}{2}$ .

$e = 0 - \frac{1}{\frac{- 4}{2}}$

Step 3.5.2.1.3

Divide $- 4$ by $2$ .

$e = 0 - \frac{1}{- 2}$

Step 3.5.2.1.4

Move the negative in front of the fraction.

$e = 0 - - \frac{1}{2}$

Step 3.5.2.1.5

Multiply $- - \frac{1}{2}$ .

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Step 3.5.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{1}{2})$

Step 3.5.2.1.5.2

Multiply $\frac{1}{2}$ by $1$ .

$e = 0 + \frac{1}{2}$

Step 3.5.2.2

Add $0$ and $\frac{1}{2}$ .

$e = \frac{1}{2}$

Step 3.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x + 1)}^{2} + \frac{1}{2}$ .

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

Step 4

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

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Step 4.1

Subtract $\frac{1}{2}$ from both sides of the equation.

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Subtract $1$ from $- 1$ .

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

Step 4.4

Divide $- 2$ by $2$ .

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

Step 5

Multiply each term in $- \frac{1}{2} \cdot {(x + 1)}^{2} = y - 1$ by $- 2$ to eliminate the fractions.

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Step 5.1

Multiply each term in $- \frac{1}{2} \cdot {(x + 1)}^{2} = y - 1$ by $- 2$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.1

Move the leading negative in $- \frac{{(x + 1)}^{2}}{2}$ into the numerator.

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

Step 5.2.2.2

Factor $2$ out of $- 2$ .

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

Step 5.2.2.3

Cancel the common factor.

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

Step 5.2.2.4

Rewrite the expression.

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

Step 5.2.3

Multiply.

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Step 5.2.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.2.3.2

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

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move $- 2$ to the left of $y$ .

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

Step 5.3.1.2

Multiply $- 1$ by $- 2$ .

${(x + 1)}^{2} = - 2 y + 2$

Step 6

Factor.

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Step 6.1

Factor $2$ out of $- 2 y + 2$ .

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Step 6.1.1

Factor $2$ out of $- 2 y$ .

${(x + 1)}^{2} = 2 (- y) + 2$

Step 6.1.2

Factor $2$ out of $2$ .

${(x + 1)}^{2} = 2 (- y) + 2 (1)$

Step 6.1.3

Factor $2$ out of $2 (- y) + 2 (1)$ .

${(x + 1)}^{2} = 2 (- y + 1)$

Step 6.2

Factor $- 1$ out of $- y$ .

${(x + 1)}^{2} = 2 (- (y) + 1)$

Step 6.3

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

${(x + 1)}^{2} = 2 (- (y) - 1 (- 1))$

Step 6.4

Factor $- 1$ out of $- (y) - 1 (- 1)$ .

${(x + 1)}^{2} = 2 (- (y - 1))$

Step 6.5

Multiply $- 1$ by $2$ .

${(x + 1)}^{2} = - 2 (y - 1)$