Find the Vertex p(x)=3x^2+12x+5

$p (x) = 3 x^{2} + 12 x + 5$

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} + 12 x + 5$ .

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

$a = 3$

$b = 12$

$c = 5$

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

$e = 5 - \frac{12^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

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

$e = 5 - \frac{144}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 5 - \frac{144}{12}$

Divide $144$ by $12$ .

$e = 5 - 1 \cdot 12$

Multiply $- 1$ by $12$ .

$e = 5 - 12$

Subtract $12$ from $5$ .

$e = - 7$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x + 2)}^{2} - 7$ .

$3 {(x + 2)}^{2} - 7$

Set $y$ equal to the new right side.

$y = 3 {(x + 2)}^{2} - 7$

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

$a = 3$

$h = - 2$

$k = - 7$

Find the vertex $(h, k)$ .

$(- 2, - 7)$

Find the Vertex p(x)=3x^2+12x+5

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