Find the Vertex -1/6x^2+200x

$- \frac{1}{6} x^{2} + 200 x$

Step 1

Rewrite the equation in vertex form.

Combine $x^{2}$ and $\frac{1}{6}$ .

$y = - \frac{x^{2}}{6} + 200 x$

Complete the square for $- \frac{x^{2}}{6} + 200 x$ .

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

$a = - \frac{1}{6}$

$b = 200$

$c = 0$

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

$d = \frac{200}{2 (- \frac{1}{6})}$

Simplify the right side.

Cancel the common factor of $200$ and $2$ .

Factor $2$ out of $200$ .

$d = \frac{2 \cdot 100}{2 (- \frac{1}{6})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 100}{2 (- \frac{1}{6})}$

Rewrite the expression.

$d = \frac{100}{- \frac{1}{6}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 100 (- 1 \cdot 6)$

Multiply $100 (- 1 \cdot 6)$ .

Multiply $- 1$ by $6$ .

$d = 100 \cdot - 6$

Multiply $100$ by $- 6$ .

$d = - 600$

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

$e = 0 - \frac{200^{2}}{4 (- \frac{1}{6})}$

Simplify the right side.

Simplify each term.

Raise $200$ to the power of $2$ .

$e = 0 - \frac{40000}{4 (- \frac{1}{6})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = 0 - \frac{40000}{- 4 (\frac{1}{6})}$

Combine $- 4$ and $\frac{1}{6}$ .

$e = 0 - \frac{40000}{\frac{- 4}{6}}$

Reduce the expression by cancelling the common factors.

Cancel the common factor of $- 4$ and $6$ .

Factor $2$ out of $- 4$ .

$e = 0 - \frac{40000}{\frac{2 (- 2)}{6}}$

Cancel the common factors.

Factor $2$ out of $6$ .

$e = 0 - \frac{40000}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Cancel the common factor.

$e = 0 - \frac{40000}{\frac{2 \cdot - 2}{2 \cdot 3}}$

Rewrite the expression.

$e = 0 - \frac{40000}{\frac{- 2}{3}}$

Move the negative in front of the fraction.

$e = 0 - \frac{40000}{- \frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (40000 (- \frac{3}{2}))$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$e = 0 - (40000 (\frac{- 3}{2}))$

Factor $2$ out of $40000$ .

$e = 0 - (2 (20000) \frac{- 3}{2})$

Cancel the common factor.

$e = 0 - (2 \cdot 20000 \frac{- 3}{2})$

Rewrite the expression.

$e = 0 - (20000 \cdot - 3)$

Multiply $20000$ by $- 3$ .

$e = 0 - - 60000$

Multiply $- 1$ by $- 60000$ .

$e = 0 + 60000$

Add $0$ and $60000$ .

$e = 60000$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{6} {(x - 600)}^{2} + 60000$ .

$- \frac{1}{6} {(x - 600)}^{2} + 60000$

Set $y$ equal to the new right side.

$y = - \frac{1}{6} \cdot {(x - 600)}^{2} + 60000$

Step 2

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

$a = - \frac{1}{6}$

$h = 600$

$k = 60000$

Step 3

Find the vertex $(h, k)$ .

$(600, 60000)$

Step 4

Find the Vertex -1/6x^2+200x

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