Find the Vertex Form y=x^2+3x+5/2

$y = x^{2} + 3 x + \frac{5}{2}$

Step 1

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

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

$a = 1$

$b = 3$

$c = \frac{5}{2}$

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

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

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

$e = \frac{5}{2} - \frac{3^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

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

$e = \frac{5}{2} - \frac{9}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = \frac{5}{2} - \frac{9}{4}$

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = \frac{5}{2} \cdot \frac{2}{2} - \frac{9}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{2}$ by $\frac{2}{2}$ .

$e = \frac{5 \cdot 2}{2 \cdot 2} - \frac{9}{4}$

Multiply $2$ by $2$ .

$e = \frac{5 \cdot 2}{4} - \frac{9}{4}$

Combine the numerators over the common denominator.

$e = \frac{5 \cdot 2 - 9}{4}$

Simplify the numerator.

Multiply $5$ by $2$ .

$e = \frac{10 - 9}{4}$

Subtract $9$ from $10$ .

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

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{3}{2})}^{2} + \frac{1}{4}$ .

${(x + \frac{3}{2})}^{2} + \frac{1}{4}$

Step 2

Set $y$ equal to the new right side.

$y = {(x + \frac{3}{2})}^{2} + \frac{1}{4}$