Find the Standard Form of the Parabola x^2-6x-8y+17=0

$x^{2} - 6 x - 8 y + 17 = 0$

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

Add $8 y$ to both sides of the equation.

$x^{2} - 6 x + 17 = 8 y$

Subtract $17$ from both sides of the equation.

$x^{2} - 6 x = 8 y - 17$

Complete the square.

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

$a = 1$

$b = - 6$

$c = 0 = 8 y - 17$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = 8 y - 17$

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

$e = 0 - \frac{{(- 6)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{36}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{36}{4}$

Divide $36$ by $4$ .

$e = 0 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Subtract $9$ from $0$ .

$e = - 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 3)}^{2} - 9$ .

${(x - 3)}^{2} - 9 = 8 y - 17$

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

Add $9$ to both sides of the equation.

${(x - 3)}^{2} = 8 y - 17 + 9$

Add $- 17$ and $9$ .

${(x - 3)}^{2} = 8 y - 8$

Factor $8$ out of $8 y - 8$ .

Factor $8$ out of $8 y$ .

${(x - 3)}^{2} = 8 (y) - 8$

Factor $8$ out of $- 8$ .

${(x - 3)}^{2} = 8 (y) + 8 (- 1)$

Factor $8$ out of $8 (y) + 8 (- 1)$ .

${(x - 3)}^{2} = 8 (y - 1)$

Find the Standard Form of the Parabola x^2-6x-8y+17=0

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