Find the Vertex (x^2)/2+4x+6

$\frac{x^{2}}{2} + 4 x + 6$

Step 1

Rewrite the equation in vertex form.

Complete the square for $\frac{x^{2}}{2} + 4 x + 6$ .

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

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

$b = 4$

$c = 6$

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

$d = \frac{4}{2 (\frac{1}{2})}$

Simplify the right side.

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

$d = 2 \cdot 2$

Multiply $2$ by $2$ .

$d = 4$

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $4^{2}$ and $4$ .

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

$e = 6 - \frac{4}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 6 - (4 \cdot 2)$

Multiply $- (4 \cdot 2)$ .

Multiply $4$ by $2$ .

$e = 6 - 1 \cdot 8$

Multiply $- 1$ by $8$ .

$e = 6 - 8$

Subtract $8$ from $6$ .

$e = - 2$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = - 4$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(- 4, - 2)$

Step 4

Find the Vertex (x^2)/2+4x+6

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