Find the Vertex (x^2)/2+6x+22

$\frac{x^{2}}{2} + 6 x + 22$

Step 1

Rewrite the equation in vertex form.

Complete the square for $\frac{x^{2}}{2} + 6 x + 22$ .

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

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

$b = 6$

$c = 22$

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

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

Simplify the right side.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 (\frac{1}{2})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 (\frac{1}{2})}$

Rewrite the expression.

$d = \frac{3}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 3 \cdot 2$

Multiply $3$ by $2$ .

$d = 6$

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

$e = 22 - \frac{6^{2}}{4 (\frac{1}{2})}$

Simplify the right side.

Simplify each term.

Raise $6$ to the power of $2$ .

$e = 22 - \frac{36}{4 (\frac{1}{2})}$

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

$e = 22 - \frac{36}{\frac{4}{2}}$

Divide $4$ by $2$ .

$e = 22 - \frac{36}{2}$

Divide $36$ by $2$ .

$e = 22 - 1 \cdot 18$

Multiply $- 1$ by $18$ .

$e = 22 - 18$

Subtract $18$ from $22$ .

$e = 4$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x + 6)}^{2} + 4$ .

$\frac{1}{2} {(x + 6)}^{2} + 4$

Set $y$ equal to the new right side.

$y = \frac{1}{2} \cdot {(x + 6)}^{2} + 4$

Step 2

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

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

$h = - 6$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(- 6, 4)$

Step 4

Find the Vertex (x^2)/2+6x+22

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