Find the Vertex y=x^2-20x+105

$y = x^{2} - 20 x + 105$

Step 1

Rewrite the equation in vertex form.

Complete the square for $x^{2} - 20 x + 105$ .

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

$a = 1$

$b = - 20$

$c = 105$

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

$d = \frac{- 20}{2 \cdot 1}$

Cancel the common factor of $- 20$ and $2$ .

Factor $2$ out of $- 20$ .

$d = \frac{2 \cdot - 10}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 10}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 10}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{- 10}{1}$

Divide $- 10$ by $1$ .

$d = - 10$

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

$e = 105 - \frac{{(- 20)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 20$ to the power of $2$ .

$e = 105 - \frac{400}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 105 - \frac{400}{4}$

Divide $400$ by $4$ .

$e = 105 - 1 \cdot 100$

Multiply $- 1$ by $100$ .

$e = 105 - 100$

Subtract $100$ from $105$ .

$e = 5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 10)}^{2} + 5$ .

${(x - 10)}^{2} + 5$

Set $y$ equal to the new right side.

$y = {(x - 10)}^{2} + 5$

Step 2

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

$a = 1$

$h = 10$

$k = 5$

Step 3

Find the vertex $(h, k)$ .

$(10, 5)$

Step 4

Find the Vertex y=x^2-20x+105

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