Algebra Examples

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

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

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

Add $1$ to both sides of the equation.

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

Complete the square.

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

$a = 2$

$b = - 2$

$c = 0 = y + 1$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{- 2}{2 \cdot 2}$

Simplify the right side.

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Cancel the common factor of $- 2$ and $2$ .

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Factor $2$ out of $- 2$ .

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

Cancel the common factors.

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Factor $2$ out of $2 \cdot 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

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

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

Simplify the right side.

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Simplify each term.

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Cancel the common factor of ${(- 2)}^{2}$ and $2$ .

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Rewrite $- 2$ as $- 1 (2)$ .

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

Apply the product rule to $- 1 (2)$ .

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

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

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

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

$e = 0 - \frac{2^{2}}{4 \cdot 2}$

Factor $2$ out of $2^{2}$ .

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

Cancel the common factors.

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Factor $2$ out of $4 \cdot 2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Subtract $\frac{1}{2}$ from $0$ .

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

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

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

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

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Add $\frac{1}{2}$ to both sides of the equation.

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Divide each term in $2 {(x - \frac{1}{2})}^{2} = y + \frac{3}{2}$ by $2$ and simplify.

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Divide each term in $2 {(x - \frac{1}{2})}^{2} = y + \frac{3}{2}$ by $2$ .

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

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

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

Simplify the right side.

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Simplify each term.

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Multiply the numerator by the reciprocal of the denominator.

${(x - \frac{1}{2})}^{2} = \frac{y}{2} + \frac{3}{2} \cdot \frac{1}{2}$

Multiply $\frac{3}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{3}{2}$ by $\frac{1}{2}$ .

${(x - \frac{1}{2})}^{2} = \frac{y}{2} + \frac{3}{2 \cdot 2}$

Multiply $2$ by $2$ .

${(x - \frac{1}{2})}^{2} = \frac{y}{2} + \frac{3}{4}$

Factor.

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To write $\frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x - \frac{1}{2})}^{2} = \frac{y}{2} \cdot \frac{2}{2} + \frac{3}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

${(x - \frac{1}{2})}^{2} = \frac{y \cdot 2}{2 \cdot 2} + \frac{3}{4}$

Multiply $2$ by $2$ .

${(x - \frac{1}{2})}^{2} = \frac{y \cdot 2}{4} + \frac{3}{4}$

Combine the numerators over the common denominator.

${(x - \frac{1}{2})}^{2} = \frac{y \cdot 2 + 3}{4}$

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

${(x - \frac{1}{2})}^{2} = \frac{2 y + 3}{4}$

Reorder terms.

${(x - \frac{1}{2})}^{2} = \frac{1}{4} (2 y + 3)$