Find the Vertex Form y=-3x^2-18x-31

$y = - 3 x^{2} - 18 x - 31$

Step 1

Complete the square for $- 3 x^{2} - 18 x - 31$ .

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

$a = - 3$

$b = - 18$

$c = - 31$

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

$d = \frac{- 18}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $- 18$ and $2$ .

Factor $2$ out of $- 18$ .

$d = \frac{2 \cdot - 9}{2 \cdot - 3}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 9}{2 (- 3)}$

Cancel the common factor.

$d = \frac{2 \cdot - 9}{2 \cdot - 3}$

Rewrite the expression.

$d = \frac{- 9}{- 3}$

Cancel the common factor of $- 9$ and $- 3$ .

Factor $- 3$ out of $- 9$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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

$e = - 31 - \frac{{(- 18)}^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Raise $- 18$ to the power of $2$ .

$e = - 31 - \frac{324}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = - 31 - \frac{324}{- 12}$

Divide $324$ by $- 12$ .

$e = - 31 - - 27$

Multiply $- 1$ by $- 27$ .

$e = - 31 + 27$

Add $- 31$ and $27$ .

$e = - 4$

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

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

Step 2

Set $y$ equal to the new right side.

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

Find the Vertex Form y=-3x^2-18x-31

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