Find the Focus x^2-6x-y+11=0

$x^{2} - 6 x - y + 11 = 0$

Step 1

Rewrite the equation in vertex form.

Isolate $y$ to the left side of the equation.

Move all terms not containing $y$ to the right side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$- 6 x - y + 11 = - x^{2}$

Add $6 x$ to both sides of the equation.

$- y + 11 = - x^{2} + 6 x$

Subtract $11$ from both sides of the equation.

$- y = - x^{2} + 6 x - 11$

Divide each term in $- y = - x^{2} + 6 x - 11$ by $- 1$ and simplify.

Divide each term in $- y = - x^{2} + 6 x - 11$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- x^{2}}{- 1} + \frac{6 x}{- 1} + \frac{- 11}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- x^{2}}{- 1} + \frac{6 x}{- 1} + \frac{- 11}{- 1}$

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{- 1} + \frac{6 x}{- 1} + \frac{- 11}{- 1}$

Simplify the right side.

Simplify each term.

Dividing two negative values results in a positive value.

$y = \frac{x^{2}}{1} + \frac{6 x}{- 1} + \frac{- 11}{- 1}$

Divide $x^{2}$ by $1$ .

$y = x^{2} + \frac{6 x}{- 1} + \frac{- 11}{- 1}$

Move the negative one from the denominator of $\frac{6 x}{- 1}$ .

$y = x^{2} - 1 \cdot (6 x) + \frac{- 11}{- 1}$

Rewrite $- 1 \cdot (6 x)$ as $- (6 x)$ .

$y = x^{2} - (6 x) + \frac{- 11}{- 1}$

Multiply $6$ by $- 1$ .

$y = x^{2} - 6 x + \frac{- 11}{- 1}$

Divide $- 11$ by $- 1$ .

$y = x^{2} - 6 x + 11$

Complete the square for $x^{2} - 6 x + 11$ .

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

$a = 1$

$b = - 6$

$c = 11$

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

$d = \frac{- 6}{2 \cdot 1}$

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

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

Cancel the common factors.

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

$d = \frac{2 \cdot - 3}{2 (1)}$

Cancel the common factor.

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

Rewrite the expression.

$d = \frac{- 3}{1}$

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = 11 - \frac{{(- 6)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$e = 11 - \frac{36}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 11 - \frac{36}{4}$

Divide $36$ by $4$ .

$e = 11 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 11 - 9$

Subtract $9$ from $11$ .

$e = 2$

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

${(x - 3)}^{2} + 2$

Set $y$ equal to the new right side.

$y = {(x - 3)}^{2} + 2$

Step 2

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

$a = 1$

$h = 3$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(3, 2)$

Step 4

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot 1}$

Cancel the common factor of $1$ .

Cancel the common factor.

$\frac{1}{4 \cdot 1}$

Rewrite the expression.

$\frac{1}{4}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(3, \frac{9}{4})$

Step 6

Find the Focus x^2-6x-y+11=0

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