Find the Vertex -2.5x^2+500x

$- 2.5 x^{2} + 500 x$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 2.5 x^{2} + 500 x$ .

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

$a = - 2.5$

$b = 500$

$c = 0$

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

$d = \frac{500}{2 \cdot - 2.5}$

Simplify the right side.

Cancel the common factor of $500$ and $2$ .

Factor $2$ out of $500$ .

$d = \frac{2 \cdot 250}{2 \cdot - 2.5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 2.5$ .

$d = \frac{2 \cdot 250}{2 (- 2.5)}$

Cancel the common factor.

$d = \frac{2 \cdot 250}{2 \cdot - 2.5}$

Rewrite the expression.

$d = \frac{250}{- 2.5}$

Divide $250$ by $- 2.5$ .

$d = - 100$

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

$e = 0 - \frac{500^{2}}{4 \cdot - 2.5}$

Simplify the right side.

Simplify each term.

Raise $500$ to the power of $2$ .

$e = 0 - \frac{250000}{4 \cdot - 2.5}$

Multiply $4$ by $- 2.5$ .

$e = 0 - \frac{250000}{- 10}$

Divide $250000$ by $- 10$ .

$e = 0 - - 25000$

Multiply $- 1$ by $- 25000$ .

$e = 0 + 25000$

Add $0$ and $25000$ .

$e = 25000$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 2.5 {(x - 100)}^{2} + 25000$ .

$- 2.5 {(x - 100)}^{2} + 25000$

Set $y$ equal to the new right side.

$y = - 2.5 {(x - 100)}^{2} + 25000$

Step 2

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

$a = - 2.5$

$h = 100$

$k = 25000$

Step 3

Find the vertex $(h, k)$ .

$(100, 25000)$

Step 4

Find the Vertex -2.5x^2+500x

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