Find the Vertex p(x)=3x^2+6x+1

$p (x) = 3 x^{2} + 6 x + 1$

Step 1

Rewrite the equation in vertex form.

Complete the square for $3 x^{2} + 6 x + 1$ .

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

$a = 3$

$b = 6$

$c = 1$

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

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

Simplify the right side.

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

$e = 1 - \frac{6^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $6$ to the power of $2$ .

$e = 1 - \frac{36}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 1 - \frac{36}{12}$

Divide $36$ by $12$ .

$e = 1 - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$e = 1 - 3$

Subtract $3$ from $1$ .

$e = - 2$

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

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

Set $y$ equal to the new right side.

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

Step 2

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

$a = 3$

$h = - 1$

$k = - 2$

Step 3

Find the vertex $(h, k)$ .

$(- 1, - 2)$

Step 4

Find the Vertex p(x)=3x^2+6x+1

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