Find the Vertex p(x)=-750x^2+15000x

$p (x) = - 750 x^{2} + 15000 x$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 750 x^{2} + 15000 x$ .

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

$a = - 750$

$b = 15000$

$c = 0$

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

$d = \frac{15000}{2 \cdot - 750}$

Simplify the right side.

Cancel the common factor of $15000$ and $2$ .

Factor $2$ out of $15000$ .

$d = \frac{2 \cdot 7500}{2 \cdot - 750}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 750$ .

$d = \frac{2 \cdot 7500}{2 (- 750)}$

Cancel the common factor.

$d = \frac{2 \cdot 7500}{2 \cdot - 750}$

Rewrite the expression.

$d = \frac{7500}{- 750}$

Cancel the common factor of $7500$ and $- 750$ .

Factor $750$ out of $7500$ .

$d = \frac{750 (10)}{- 750}$

Move the negative one from the denominator of $\frac{10}{- 1}$ .

$d = - 1 \cdot 10$

Multiply $- 1$ by $10$ .

$d = - 10$

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

$e = 0 - \frac{15000^{2}}{4 \cdot - 750}$

Simplify the right side.

Simplify each term.

Raise $15000$ to the power of $2$ .

$e = 0 - \frac{225000000}{4 \cdot - 750}$

Multiply $4$ by $- 750$ .

$e = 0 - \frac{225000000}{- 3000}$

Divide $225000000$ by $- 3000$ .

$e = 0 - - 75000$

Multiply $- 1$ by $- 75000$ .

$e = 0 + 75000$

Add $0$ and $75000$ .

$e = 75000$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 750 {(x - 10)}^{2} + 75000$ .

$- 750 {(x - 10)}^{2} + 75000$

Set $y$ equal to the new right side.

$y = - 750 {(x - 10)}^{2} + 75000$

Step 2

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

$a = - 750$

$h = 10$

$k = 75000$

Step 3

Find the vertex $(h, k)$ .

$(10, 75000)$

Step 4

Find the Vertex p(x)=-750x^2+15000x

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