Find the Vertex y=x^2-18x+81

$y = x^{2} - 18 x + 81$

Step 1

Rewrite the equation in vertex form.

Complete the square for $x^{2} - 18 x + 81$ .

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

$a = 1$

$b = - 18$

$c = 81$

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

$d = \frac{- 18}{2 \cdot 1}$

Cancel the common factor of $- 18$ and $2$ .

Factor $2$ out of $- 18$ .

$d = \frac{2 \cdot - 9}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 9}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 9}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 9}{1}$

Divide $- 9$ by $1$ .

$d = - 9$

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

$e = 81 - \frac{{(- 18)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 18$ to the power of $2$ .

$e = 81 - \frac{324}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 81 - \frac{324}{4}$

Divide $324$ by $4$ .

$e = 81 - 1 \cdot 81$

Multiply $- 1$ by $81$ .

$e = 81 - 81$

Subtract $81$ from $81$ .

$e = 0$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 9)}^{2} + 0$ .

${(x - 9)}^{2} + 0$

Set $y$ equal to the new right side.

$y = {(x - 9)}^{2} + 0$

Step 2

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

$a = 1$

$h = 9$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(9, 0)$

Step 4

Find the Vertex y=x^2-18x+81

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