Find the Vertex 2x^2+20x+51

$2 x^{2} + 20 x + 51$

Step 1

Rewrite the equation in vertex form.

Complete the square for $2 x^{2} + 20 x + 51$ .

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

$a = 2$

$b = 20$

$c = 51$

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

$d = \frac{20}{2 \cdot 2}$

Simplify the right side.

Cancel the common factor of $20$ and $2$ .

Factor $2$ out of $20$ .

$d = \frac{2 \cdot 10}{2 \cdot 2}$

Cancel the common factors.

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

$d = \frac{2 \cdot 10}{2 (2)}$

Cancel the common factor.

$d = \frac{2 \cdot 10}{2 \cdot 2}$

Rewrite the expression.

$d = \frac{10}{2}$

Cancel the common factor of $10$ and $2$ .

Factor $2$ out of $10$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

$d = \frac{2 \cdot 5}{2 \cdot 1}$

Rewrite the expression.

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

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

$e = 51 - \frac{20^{2}}{4 \cdot 2}$

Simplify the right side.

Simplify each term.

Raise $20$ to the power of $2$ .

$e = 51 - \frac{400}{4 \cdot 2}$

Multiply $4$ by $2$ .

$e = 51 - \frac{400}{8}$

Divide $400$ by $8$ .

$e = 51 - 1 \cdot 50$

Multiply $- 1$ by $50$ .

$e = 51 - 50$

Subtract $50$ from $51$ .

$e = 1$

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

$2 {(x + 5)}^{2} + 1$

Set $y$ equal to the new right side.

$y = 2 {(x + 5)}^{2} + 1$

Step 2

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

$a = 2$

$h = - 5$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(- 5, 1)$

Step 4

Find the Vertex 2x^2+20x+51

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