Find the Vertex -12c^2+120c

$- 12 c^{2} + 120 c$

Step 1

Rewrite the equation in terms of $x$ and $y$ .

$y = - 12 x^{2} + 120 x$

Step 2

Rewrite the equation in vertex form.

Complete the square for $- 12 x^{2} + 120 x$ .

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

$a = - 12$

$b = 120$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{120}{2 \cdot - 12}$

Simplify the right side.

Cancel the common factor of $120$ and $2$ .

Factor $2$ out of $120$ .

$d = \frac{2 \cdot 60}{2 \cdot - 12}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 12$ .

$d = \frac{2 \cdot 60}{2 (- 12)}$

Cancel the common factor.

$d = \frac{2 \cdot 60}{2 \cdot - 12}$

Rewrite the expression.

$d = \frac{60}{- 12}$

Cancel the common factor of $60$ and $- 12$ .

Factor $12$ out of $60$ .

$d = \frac{12 (5)}{- 12}$

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$d = - 5$

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

$e = 0 - \frac{120^{2}}{4 \cdot - 12}$

Simplify the right side.

Simplify each term.

Raise $120$ to the power of $2$ .

$e = 0 - \frac{14400}{4 \cdot - 12}$

Multiply $4$ by $- 12$ .

$e = 0 - \frac{14400}{- 48}$

Divide $14400$ by $- 48$ .

$e = 0 - - 300$

Multiply $- 1$ by $- 300$ .

$e = 0 + 300$

Add $0$ and $300$ .

$e = 300$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 12 {(x - 5)}^{2} + 300$ .

$- 12 {(x - 5)}^{2} + 300$

Set $y$ equal to the new right side.

$y = - 12 {(x - 5)}^{2} + 300$

Step 3

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 12$

$h = 5$

$k = 300$

Step 4

Find the vertex $(h, k)$ .

$(5, 300)$

Step 5

Find the Vertex -12c^2+120c

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