Find the Vertex Form y=-3x^2-24x-46

$y = - 3 x^{2} - 24 x - 46$

Step 1

Complete the square for $- 3 x^{2} - 24 x - 46$ .

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

$a = - 3$

$b = - 24$

$c = - 46$

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

$d = \frac{- 24}{2 \cdot - 3}$

Simplify the right side.

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

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot - 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

$d = \frac{2 \cdot - 12}{2 (- 3)}$

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot - 3}$

Rewrite the expression.

$d = \frac{- 12}{- 3}$

Cancel the common factor of $- 12$ and $- 3$ .

Factor $- 3$ out of $- 12$ .

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

Cancel the common factors.

Factor $- 3$ out of $- 3$ .

$d = \frac{- 3 \cdot 4}{- 3 \cdot 1}$

Cancel the common factor.

$d = \frac{- 3 \cdot 4}{- 3 \cdot 1}$

Rewrite the expression.

$d = \frac{4}{1}$

Divide $4$ by $1$ .

$d = 4$

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

$e = - 46 - \frac{{(- 24)}^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

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

$e = - 46 - \frac{576}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = - 46 - \frac{576}{- 12}$

Divide $576$ by $- 12$ .

$e = - 46 - - 48$

Multiply $- 1$ by $- 48$ .

$e = - 46 + 48$

Add $- 46$ and $48$ .

$e = 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x + 4)}^{2} + 2$ .

$- 3 {(x + 4)}^{2} + 2$

Step 2

Set $y$ equal to the new right side.

$y = - 3 {(x + 4)}^{2} + 2$

Find the Vertex Form y=-3x^2-24x-46

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