Find the Vertex -1/2x^2+7x-5

$- \frac{1}{2} x^{2} + 7 x - 5$

Step 1

Rewrite the equation in vertex form.

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

$y = - \frac{x^{2}}{2} + 7 x - 5$

Complete the square for $- \frac{x^{2}}{2} + 7 x - 5$ .

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

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

$b = 7$

$c = - 5$

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

$d = \frac{7}{2 (- \frac{1}{2})}$

Simplify the right side.

Simplify the denominator.

Multiply $- 1$ by $2$ .

$d = \frac{7}{- 2 (\frac{1}{2})}$

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

$d = \frac{7}{\frac{- 2}{2}}$

Divide $- 2$ by $2$ .

$d = \frac{7}{- 1}$

Divide $7$ by $- 1$ .

$d = - 7$

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

$e = - 5 - \frac{7^{2}}{4 (- \frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $7$ to the power of $2$ .

$e = - 5 - \frac{49}{4 (- \frac{1}{2})}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$e = - 5 - \frac{49}{- 4 (\frac{1}{2})}$

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

$e = - 5 - \frac{49}{\frac{- 4}{2}}$

Divide $- 4$ by $2$ .

$e = - 5 - \frac{49}{- 2}$

Move the negative in front of the fraction.

$e = - 5 - - \frac{49}{2}$

Multiply $- - \frac{49}{2}$ .

Multiply $- 1$ by $- 1$ .

$e = - 5 + 1 (\frac{49}{2})$

Multiply $\frac{49}{2}$ by $1$ .

$e = - 5 + \frac{49}{2}$

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - 5 \cdot \frac{2}{2} + \frac{49}{2}$

Combine $- 5$ and $\frac{2}{2}$ .

$e = \frac{- 5 \cdot 2}{2} + \frac{49}{2}$

Combine the numerators over the common denominator.

$e = \frac{- 5 \cdot 2 + 49}{2}$

Simplify the numerator.

Multiply $- 5$ by $2$ .

$e = \frac{- 10 + 49}{2}$

Add $- 10$ and $49$ .

$e = \frac{39}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - 7)}^{2} + \frac{39}{2}$ .

$- \frac{1}{2} {(x - 7)}^{2} + \frac{39}{2}$

Set $y$ equal to the new right side.

$y = - \frac{1}{2} \cdot {(x - 7)}^{2} + \frac{39}{2}$

Step 2

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

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

$h = 7$

$k = \frac{39}{2}$

Step 3

Find the vertex $(h, k)$ .

$(7, \frac{39}{2})$

Step 4

Find the Vertex -1/2x^2+7x-5

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