Find the Vertex Form y=3x^2+12x+23/2

$y = 3 x^{2} + 12 x + \frac{23}{2}$

Complete the square for $3 x^{2} + 12 x + \frac{23}{2}$ .

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

$a = 3$

$b = 12$

$c = \frac{23}{2}$

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

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

Simplify the right side.

Cancel the common factor of $12$ and $2$ .

Factor $2$ out of $12$ .

$d = \frac{2 \cdot 6}{2 \cdot 3}$

Cancel the common factors.

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

$d = \frac{2 \cdot 6}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot 6}{2 \cdot 3}$

Rewrite the expression.

$d = \frac{6}{3}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

$d = \frac{3 \cdot 2}{3 (1)}$

Cancel the common factor.

$d = \frac{3 \cdot 2}{3 \cdot 1}$

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

$e = \frac{23}{2} - \frac{12^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $12$ to the power of $2$ .

$e = \frac{23}{2} - \frac{144}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = \frac{23}{2} - \frac{144}{12}$

Divide $144$ by $12$ .

$e = \frac{23}{2} - 1 \cdot 12$

Multiply $- 1$ by $12$ .

$e = \frac{23}{2} - 12$

To write $- 12$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = \frac{23}{2} - 12 \cdot \frac{2}{2}$

Combine $- 12$ and $\frac{2}{2}$ .

$e = \frac{23}{2} + \frac{- 12 \cdot 2}{2}$

Combine the numerators over the common denominator.

$e = \frac{23 - 12 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 12$ by $2$ .

$e = \frac{23 - 24}{2}$

Subtract $24$ from $23$ .

$e = \frac{- 1}{2}$

Move the negative in front of the fraction.

$e = - \frac{1}{2}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 2)}^{2} - \frac{1}{2}$ .

$3 {(x + 2)}^{2} - \frac{1}{2}$

Set $y$ equal to the new right side.

$y = 3 {(x + 2)}^{2} - \frac{1}{2}$

Find the Vertex Form y=3x^2+12x+23/2

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