Find the Vertex g(x)=2x^2+36x+170

$g (x) = 2 x^{2} + 36 x + 170$

Step 1

Rewrite the equation in vertex form.

Complete the square for $2 x^{2} + 36 x + 170$ .

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

$a = 2$

$b = 36$

$c = 170$

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

$d = \frac{36}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $36$ and $2$ .

Factor $2$ out of $36$ .

$d = \frac{2 \cdot 18}{2 \cdot 2}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 2$ .

$d = \frac{2 \cdot 18}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot 18}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{18}{2}$

Cancel the common factor of $18$ and $2$ .

Factor $2$ out of $18$ .

$d = \frac{2 \cdot 9}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$d = \frac{2 \cdot 9}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 9}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{9}{1}$

Divide $9$ by $1$ .

$d = 9$

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

$e = 170 - \frac{36^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $36$ to the power of $2$ .

$e = 170 - \frac{1296}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 170 - \frac{1296}{8}$

Divide $1296$ by $8$ .

$e = 170 - 1 \cdot 162$

Multiply $- 1$ by $162$ .

$e = 170 - 162$

Subtract $162$ from $170$ .

$e = 8$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + 9)}^{2} + 8$ .

$2 {(x + 9)}^{2} + 8$

Set $y$ equal to the new right side.

$y = 2 {(x + 9)}^{2} + 8$

Step 2

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

$a = 2$

$h = - 9$

$k = 8$

Step 3

Find the vertex $(h, k)$ .

$(- 9, 8)$

Step 4

Find the Vertex g(x)=2x^2+36x+170

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