Find the Vertex -8x^2-24x+3

$- 8 x^{2} - 24 x + 3$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 8 x^{2} - 24 x + 3$ .

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

$a = - 8$

$b = - 24$

$c = 3$

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

$d = \frac{- 24}{2 \cdot - 8}$

Simplify the right side.

Cancel the common factor of $- 24$ and $2$ .

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot - 8}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 12}{2 (- 8)}$

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot - 8}$

Rewrite the expression.

$d = \frac{- 12}{- 8}$

Cancel the common factor of $- 12$ and $- 8$ .

Factor $- 4$ out of $- 12$ .

$d = \frac{- 4 (3)}{- 8}$

Cancel the common factors.

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

$d = \frac{- 4 \cdot 3}{- 4 \cdot 2}$

Cancel the common factor.

$d = \frac{- 4 \cdot 3}{- 4 \cdot 2}$

Rewrite the expression.

$d = \frac{3}{2}$

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

$e = 3 - \frac{{(- 24)}^{2}}{4 \cdot - 8}$

Simplify the right side.

Simplify each term.

Raise $- 24$ to the power of $2$ .

$e = 3 - \frac{576}{4 \cdot - 8}$

Multiply $4$ by $- 8$ .

$e = 3 - \frac{576}{- 32}$

Divide $576$ by $- 32$ .

$e = 3 - - 18$

Multiply $- 1$ by $- 18$ .

$e = 3 + 18$

Add $3$ and $18$ .

$e = 21$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 8 {(x + \frac{3}{2})}^{2} + 21$ .

$- 8 {(x + \frac{3}{2})}^{2} + 21$

Set $y$ equal to the new right side.

$y = - 8 {(x + \frac{3}{2})}^{2} + 21$

Step 2

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

$a = - 8$

$h = - \frac{3}{2}$

$k = 21$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{3}{2}, 21)$

Step 4

Find the Vertex -8x^2-24x+3

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