Find the Vertex f(x)=x^2+2x+5

$f (x) = x^{2} + 2 x + 5$

Rewrite the equation in vertex form.

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

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

$a = 1$

$b = 2$

$c = 5$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $1$ .

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

Cancel the common factor of $4$ .

Cancel the common factor.

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

Rewrite the expression.

$e = 5 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = 5 - 1$

Subtract $1$ from $5$ .

$e = 4$

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

${(x + 1)}^{2} + 4$

Set $y$ equal to the new right side.

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

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

$a = 1$

$h = - 1$

$k = 4$

Find the vertex $(h, k)$ .

$(- 1, 4)$