Find the Standard Form of the Parabola y=6x^2-48x+100

$y = 6 x^{2} - 48 x + 100$

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.

$6 x^{2} - 48 x + 100 = y$

Step 2

Subtract $100$ from both sides of the equation.

$6 x^{2} - 48 x = y - 100$

Step 3

Complete the square.

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

$a = 6$

$b = - 48$

$c = 0 = y - 100$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 100$

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

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

$d = \frac{- 48}{2 \cdot 6}$

Simplify the right side.

Cancel the common factor of $- 48$ and $2$ .

Factor $2$ out of $- 48$ .

$d = \frac{2 \cdot - 24}{2 \cdot 6}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 6$ .

$d = \frac{2 \cdot - 24}{2 (6)}$

Cancel the common factor.

$d = \frac{2 \cdot - 24}{2 \cdot 6}$

Rewrite the expression.

$d = \frac{- 24}{6}$

Cancel the common factor of $- 24$ and $6$ .

Factor $6$ out of $- 24$ .

$d = \frac{6 \cdot - 4}{6}$

Cancel the common factors.

Factor $6$ out of $6$ .

$d = \frac{6 \cdot - 4}{6 (1)}$

Cancel the common factor.

$d = \frac{6 \cdot - 4}{6 \cdot 1}$

Rewrite the expression.

$d = \frac{- 4}{1}$

Divide $- 4$ by $1$ .

$d = - 4$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 48)}^{2}}{4 \cdot 6}$

Simplify the right side.

Simplify each term.

Raise $- 48$ to the power of $2$ .

$e = 0 - \frac{2304}{4 \cdot 6}$

Multiply $4$ by $6$ .

$e = 0 - \frac{2304}{24}$

Divide $2304$ by $24$ .

$e = 0 - 1 \cdot 96$

Multiply $- 1$ by $96$ .

$e = 0 - 96$

Subtract $96$ from $0$ .

$e = - 96$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(x - 4)}^{2} - 96$ .

$6 {(x - 4)}^{2} - 96 = y - 100$

Step 4

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

Add $96$ to both sides of the equation.

$6 {(x - 4)}^{2} = y - 100 + 96$

Add $- 100$ and $96$ .

$6 {(x - 4)}^{2} = y - 4$

Step 5

Divide each term in $6 {(x - 4)}^{2} = y - 4$ by $6$ and simplify.

Divide each term in $6 {(x - 4)}^{2} = y - 4$ by $6$ .

$\frac{6 {(x - 4)}^{2}}{6} = \frac{y}{6} + \frac{- 4}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 {(x - 4)}^{2}}{6} = \frac{y}{6} + \frac{- 4}{6}$

Divide ${(x - 4)}^{2}$ by $1$ .

${(x - 4)}^{2} = \frac{y}{6} + \frac{- 4}{6}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $- 4$ .

${(x - 4)}^{2} = \frac{y}{6} + \frac{2 (- 2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

${(x - 4)}^{2} = \frac{y}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Cancel the common factor.

${(x - 4)}^{2} = \frac{y}{6} + \frac{2 \cdot - 2}{2 \cdot 3}$

Rewrite the expression.

${(x - 4)}^{2} = \frac{y}{6} + \frac{- 2}{3}$

Move the negative in front of the fraction.

${(x - 4)}^{2} = \frac{y}{6} - \frac{2}{3}$

Step 6

Factor.

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x - 4)}^{2} = \frac{y}{6} - \frac{2}{3} \cdot \frac{2}{2}$

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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

${(x - 4)}^{2} = \frac{y}{6} - \frac{2 \cdot 2}{3 \cdot 2}$

Multiply $3$ by $2$ .

${(x - 4)}^{2} = \frac{y}{6} - \frac{2 \cdot 2}{6}$

Combine the numerators over the common denominator.

${(x - 4)}^{2} = \frac{y - 2 \cdot 2}{6}$

Multiply $- 2$ by $2$ .

${(x - 4)}^{2} = \frac{y - 4}{6}$

Reorder terms.

${(x - 4)}^{2} = \frac{1}{6} (y - 4)$

Find the Standard Form of the Parabola y=6x^2-48x+100

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