Find the Vertex y=5x^2+20x+9

$y = 5 x^{2} + 20 x + 9$

Step 1

Rewrite the equation in vertex form.

Complete the square for $5 x^{2} + 20 x + 9$ .

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

$a = 5$

$b = 20$

$c = 9$

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

$d = \frac{20}{2 \cdot 5}$

Simplify the right side.

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

Factor $2$ out of $20$ .

$d = \frac{2 \cdot 10}{2 \cdot 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 5$ .

$d = \frac{2 \cdot 10}{2 (5)}$

Cancel the common factor.

$d = \frac{2 \cdot 10}{2 \cdot 5}$

Rewrite the expression.

$d = \frac{10}{5}$

Cancel the common factor of $10$ and $5$ .

Factor $5$ out of $10$ .

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

Cancel the common factors.

Factor $5$ out of $5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$d = 2$

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

$e = 9 - \frac{20^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

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

$e = 9 - \frac{400}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = 9 - \frac{400}{20}$

Divide $400$ by $20$ .

$e = 9 - 1 \cdot 20$

Multiply $- 1$ by $20$ .

$e = 9 - 20$

Subtract $20$ from $9$ .

$e = - 11$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x + 2)}^{2} - 11$ .

$5 {(x + 2)}^{2} - 11$

Set $y$ equal to the new right side.

$y = 5 {(x + 2)}^{2} - 11$

Step 2

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

$a = 5$

$h = - 2$

$k = - 11$

Step 3

Find the vertex $(h, k)$ .

$(- 2, - 11)$

Step 4

Find the Vertex y=5x^2+20x+9

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