Find the Vertex -8x^2-40x+9

$- 8 x^{2} - 40 x + 9$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 8 x^{2} - 40 x + 9$ .

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

$a = - 8$

$b = - 40$

$c = 9$

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

$d = \frac{- 40}{2 \cdot - 8}$

Simplify the right side.

Cancel the common factor of $- 40$ and $2$ .

Factor $2$ out of $- 40$ .

$d = \frac{2 \cdot - 20}{2 \cdot - 8}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 8$ .

$d = \frac{2 \cdot - 20}{2 (- 8)}$

Cancel the common factor.

$d = \frac{2 \cdot - 20}{2 \cdot - 8}$

Rewrite the expression.

$d = \frac{- 20}{- 8}$

Cancel the common factor of $- 20$ and $- 8$ .

Factor $- 4$ out of $- 20$ .

$d = \frac{- 4 (5)}{- 8}$

Cancel the common factors.

Factor $- 4$ out of $- 8$ .

$d = \frac{- 4 \cdot 5}{- 4 \cdot 2}$

Cancel the common factor.

$d = \frac{- 4 \cdot 5}{- 4 \cdot 2}$

Rewrite the expression.

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

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

$e = 9 - \frac{{(- 40)}^{2}}{4 \cdot - 8}$

Simplify the right side.

Simplify each term.

Raise $- 40$ to the power of $2$ .

$e = 9 - \frac{1600}{4 \cdot - 8}$

Multiply $4$ by $- 8$ .

$e = 9 - \frac{1600}{- 32}$

Divide $1600$ by $- 32$ .

$e = 9 - - 50$

Multiply $- 1$ by $- 50$ .

$e = 9 + 50$

Add $9$ and $50$ .

$e = 59$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 8 {(x + \frac{5}{2})}^{2} + 59$ .

$- 8 {(x + \frac{5}{2})}^{2} + 59$

Set $y$ equal to the new right side.

$y = - 8 {(x + \frac{5}{2})}^{2} + 59$

Step 2

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

$a = - 8$

$h = - \frac{5}{2}$

$k = 59$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{5}{2}, 59)$

Step 4

Find the Vertex -8x^2-40x+9

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