Find the Standard Form of the Parabola y=6x^2+36x+54

$y = 6 x^{2} + 36 x + 54$

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.

$6 x^{2} + 36 x + 54 = y$

Step 2

Subtract $54$ from both sides of the equation.

$6 x^{2} + 36 x = y - 54$

Step 3

Complete the square.

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

$a = 6$

$b = 36$

$c = 0 = y - 54$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 54$

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

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

$d = \frac{36}{2 \cdot 6}$

Simplify the right side.

Cancel the common factor of $36$ and $2$ .

Factor $2$ out of $36$ .

$d = \frac{2 \cdot 18}{2 \cdot 6}$

Cancel the common factors.

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

$d = \frac{2 \cdot 18}{2 (6)}$

Cancel the common factor.

$d = \frac{2 \cdot 18}{2 \cdot 6}$

Rewrite the expression.

$d = \frac{18}{6}$

Cancel the common factor of $18$ and $6$ .

Factor $6$ out of $18$ .

$d = \frac{6 \cdot 3}{6}$

Cancel the common factors.

Factor $6$ out of $6$ .

$d = \frac{6 \cdot 3}{6 (1)}$

Cancel the common factor.

$d = \frac{6 \cdot 3}{6 \cdot 1}$

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

$e = 0 - \frac{36^{2}}{4 \cdot 6}$

Simplify the right side.

Simplify each term.

Raise $36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot 6}$

Multiply $4$ by $6$ .

$e = 0 - \frac{1296}{24}$

Divide $1296$ by $24$ .

$e = 0 - 1 \cdot 54$

Multiply $- 1$ by $54$ .

$e = 0 - 54$

Subtract $54$ from $0$ .

$e = - 54$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $6 {(x + 3)}^{2} - 54$ .

$6 {(x + 3)}^{2} - 54 = y - 54$

Step 4

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

Add $54$ to both sides of the equation.

$6 {(x + 3)}^{2} = y - 54 + 54$

Combine the opposite terms in $y - 54 + 54$ .

Add $- 54$ and $54$ .

$6 {(x + 3)}^{2} = y + 0$

Add $y$ and $0$ .

$6 {(x + 3)}^{2} = y$

Step 5

Divide each term in $6 {(x + 3)}^{2} = y$ by $6$ and simplify.

Divide each term in $6 {(x + 3)}^{2} = y$ by $6$ .

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

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

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

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

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

Step 6

Reorder terms.

${(x + 3)}^{2} = \frac{1}{6} y$

Find the Standard Form of the Parabola y=6x^2+36x+54

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