Find the Vertex y=x^2+3x+1

$y = x^{2} + 3 x + 1$

Step 1

Rewrite the equation in vertex form.

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

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

$a = 1$

$b = 3$

$c = 1$

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

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

Multiply $2$ by $1$ .

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

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

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

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$e = 1 - \frac{9}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 1 - \frac{9}{4}$

Write $1$ as a fraction with a common denominator.

$e = \frac{4}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{4 - 9}{4}$

Subtract $9$ from $4$ .

$e = \frac{- 5}{4}$

Move the negative in front of the fraction.

$e = - \frac{5}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{3}{2})}^{2} - \frac{5}{4}$ .

${(x + \frac{3}{2})}^{2} - \frac{5}{4}$

Set $y$ equal to the new right side.

$y = {(x + \frac{3}{2})}^{2} - \frac{5}{4}$

Step 2

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

$a = 1$

$h = - \frac{3}{2}$

$k = - \frac{5}{4}$

Step 3

Find the vertex $(h, k)$ .

$(- \frac{3}{2}, - \frac{5}{4})$

Step 4