Find the Vertex -1/2x^2+5x-2

$- \frac{1}{2} x^{2} + 5 x - 2$

Step 1

Rewrite the equation in vertex form.

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

$y = - \frac{x^{2}}{2} + 5 x - 2$

Complete the square for $- \frac{x^{2}}{2} + 5 x - 2$ .

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

$a = - \frac{1}{2}$

$b = 5$

$c = - 2$

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

$d = \frac{5}{2 (- \frac{1}{2})}$

Simplify the right side.

Simplify the denominator.

Multiply $- 1$ by $2$ .

$d = \frac{5}{- 2 (\frac{1}{2})}$

Combine $- 2$ and $\frac{1}{2}$ .

$d = \frac{5}{\frac{- 2}{2}}$

Divide $- 2$ by $2$ .

$d = \frac{5}{- 1}$

Divide $5$ by $- 1$ .

$d = - 5$

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

$e = - 2 - \frac{5^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $5$ to the power of $2$ .

$e = - 2 - \frac{25}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 2 - \frac{25}{- 4 (\frac{1}{2})}$

Combine $- 4$ and $\frac{1}{2}$ .

$e = - 2 - \frac{25}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = - 2 - \frac{25}{- 2}$

Move the negative in front of the fraction.

$e = - 2 - - \frac{25}{2}$

Multiply $- - \frac{25}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = - 2 + 1 (\frac{25}{2})$

Multiply $\frac{25}{2}$ by $1$ .

$e = - 2 + \frac{25}{2}$

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 2 \cdot \frac{2}{2} + \frac{25}{2}$

Combine $- 2$ and $\frac{2}{2}$ .

$e = \frac{- 2 \cdot 2}{2} + \frac{25}{2}$

Combine the numerators over the common denominator.

$e = \frac{- 2 \cdot 2 + 25}{2}$

Simplify the numerator.

Multiply $- 2$ by $2$ .

$e = \frac{- 4 + 25}{2}$

Add $- 4$ and $25$ .

$e = \frac{21}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - 5)}^{2} + \frac{21}{2}$ .

$- \frac{1}{2} {(x - 5)}^{2} + \frac{21}{2}$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x - 5)}^{2} + \frac{21}{2}$

Step 2

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

$a = - \frac{1}{2}$

$h = 5$

$k = \frac{21}{2}$

Step 3

Find the vertex $(h, k)$ .

$(5, \frac{21}{2})$

Step 4

Find the Vertex -1/2x^2+5x-2

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