Find the Vertex Form -2x^2+24x-54

$- 2 x^{2} + 24 x - 54$

Step 1

Write $- 2 x^{2} + 24 x - 54$ as an equation.

$y = - 2 x^{2} + 24 x - 54$

Step 2

Complete the square for $- 2 x^{2} + 24 x - 54$ .

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

$a = - 2$

$b = 24$

$c = - 54$

Consider the vertex form of a parabola.

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

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

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

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

Simplify the right side.

Cancel the common factor of $24$ and $2$ .

Factor $2$ out of $24$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $2$ out of $12$ .

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

Move the negative one from the denominator of $\frac{6}{- 1}$ .

$d = - 1 \cdot 6$

Multiply $- 1$ by $6$ .

$d = - 6$

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

$e = - 54 - \frac{24^{2}}{4 \cdot - 2}$

Simplify the right side.

Simplify each term.

Raise $24$ to the power of $2$ .

$e = - 54 - \frac{576}{4 \cdot - 2}$

Multiply $4$ by $- 2$ .

$e = - 54 - \frac{576}{- 8}$

Divide $576$ by $- 8$ .

$e = - 54 - - 72$

Multiply $- 1$ by $- 72$ .

$e = - 54 + 72$

Add $- 54$ and $72$ .

$e = 18$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2 {(x - 6)}^{2} + 18$ .

$- 2 {(x - 6)}^{2} + 18$

Step 3

Set $y$ equal to the new right side.

$y = - 2 {(x - 6)}^{2} + 18$

Find the Vertex Form -2x^2+24x-54

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