Find the Standard Form of the Parabola y=2x^2-12x+16

$y = 2 x^{2} - 12 x + 16$

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

$2 x^{2} - 12 x + 16 = y$

Subtract $16$ from both sides of the equation.

$2 x^{2} - 12 x = y - 16$

Complete the square.

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

$a = 2$

$b = - 12$

$c = 0 = y - 16$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 16$

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

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

$d = \frac{- 12}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $- 12$ and $2$ .

Factor $2$ out of $- 12$ .

$d = \frac{2 \cdot - 6}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 6}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot - 6}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{- 6}{2}$

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$d = \frac{2 \cdot - 3}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 3}{1}$

Divide $- 3$ by $1$ .

$d = - 3$

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

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

$e = 0 - \frac{{(- 12)}^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $- 12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 0 - \frac{144}{8}$

Divide $144$ by $8$ .

$e = 0 - 1 \cdot 18$

Multiply $- 1$ by $18$ .

$e = 0 - 18$

Subtract $18$ from $0$ .

$e = - 18$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x - 3)}^{2} - 18$ .

$2 {(x - 3)}^{2} - 18 = y - 16$

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

Add $18$ to both sides of the equation.

$2 {(x - 3)}^{2} = y - 16 + 18$

Add $- 16$ and $18$ .

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide ${(x - 3)}^{2}$ by $1$ .

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

Simplify the right side.

Divide $2$ by $2$ .

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

Factor.

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

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

Combine the numerators over the common denominator.

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

Reorder terms.

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

Find the Standard Form of the Parabola y=2x^2-12x+16

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