Find the Standard Form of the Parabola y=27x^2-216x+450

$y = 27 x^{2} - 216 x + 450$

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.

$27 x^{2} - 216 x + 450 = y$

Step 2

Subtract $450$ from both sides of the equation.

$27 x^{2} - 216 x = y - 450$

Step 3

Complete the square.

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

$a = 27$

$b = - 216$

$c = 0 = y - 450$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 450$

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

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

$d = \frac{- 216}{2 \cdot 27}$

Simplify the right side.

Cancel the common factor of $- 216$ and $2$ .

Factor $2$ out of $- 216$ .

$d = \frac{2 \cdot - 108}{2 \cdot 27}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 27$ .

$d = \frac{2 \cdot - 108}{2 (27)}$

Cancel the common factor.

$d = \frac{2 \cdot - 108}{2 \cdot 27}$

Rewrite the expression.

$d = \frac{- 108}{27}$

Cancel the common factor of $- 108$ and $27$ .

Factor $27$ out of $- 108$ .

$d = \frac{27 \cdot - 4}{27}$

Cancel the common factors.

Factor $27$ out of $27$ .

$d = \frac{27 \cdot - 4}{27 (1)}$

Cancel the common factor.

$d = \frac{27 \cdot - 4}{27 \cdot 1}$

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

$e = 0 - \frac{{(- 216)}^{2}}{4 \cdot 27}$

Simplify the right side.

Simplify each term.

Raise $- 216$ to the power of $2$ .

$e = 0 - \frac{46656}{4 \cdot 27}$

Multiply $4$ by $27$ .

$e = 0 - \frac{46656}{108}$

Divide $46656$ by $108$ .

$e = 0 - 1 \cdot 432$

Multiply $- 1$ by $432$ .

$e = 0 - 432$

Subtract $432$ from $0$ .

$e = - 432$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $27 {(x - 4)}^{2} - 432$ .

$27 {(x - 4)}^{2} - 432 = y - 450$

Step 4

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

Add $432$ to both sides of the equation.

$27 {(x - 4)}^{2} = y - 450 + 432$

Add $- 450$ and $432$ .

$27 {(x - 4)}^{2} = y - 18$

Step 5

Divide each term in $27 {(x - 4)}^{2} = y - 18$ by $27$ and simplify.

Divide each term in $27 {(x - 4)}^{2} = y - 18$ by $27$ .

$\frac{27 {(x - 4)}^{2}}{27} = \frac{y}{27} + \frac{- 18}{27}$

Simplify the left side.

Cancel the common factor of $27$ .

Cancel the common factor.

$\frac{27 {(x - 4)}^{2}}{27} = \frac{y}{27} + \frac{- 18}{27}$

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

${(x - 4)}^{2} = \frac{y}{27} + \frac{- 18}{27}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 18$ and $27$ .

Factor $9$ out of $- 18$ .

${(x - 4)}^{2} = \frac{y}{27} + \frac{9 (- 2)}{27}$

Cancel the common factors.

Factor $9$ out of $27$ .

${(x - 4)}^{2} = \frac{y}{27} + \frac{9 \cdot - 2}{9 \cdot 3}$

Cancel the common factor.

${(x - 4)}^{2} = \frac{y}{27} + \frac{9 \cdot - 2}{9 \cdot 3}$

Rewrite the expression.

${(x - 4)}^{2} = \frac{y}{27} + \frac{- 2}{3}$

Move the negative in front of the fraction.

${(x - 4)}^{2} = \frac{y}{27} - \frac{2}{3}$

Step 6

Factor.

To write $- \frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

${(x - 4)}^{2} = \frac{y}{27} - \frac{2}{3} \cdot \frac{9}{9}$

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{2}{3}$ by $\frac{9}{9}$ .

${(x - 4)}^{2} = \frac{y}{27} - \frac{2 \cdot 9}{3 \cdot 9}$

Multiply $3$ by $9$ .

${(x - 4)}^{2} = \frac{y}{27} - \frac{2 \cdot 9}{27}$

Combine the numerators over the common denominator.

${(x - 4)}^{2} = \frac{y - 2 \cdot 9}{27}$

Multiply $- 2$ by $9$ .

${(x - 4)}^{2} = \frac{y - 18}{27}$

Reorder terms.

${(x - 4)}^{2} = \frac{1}{27} (y - 18)$

Find the Standard Form of the Parabola y=27x^2-216x+450

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