Question

put the equation in standard form.
x² + y² - 16x + 6y + 64 = 0
(x - ?)² + (y + )² =

Explanation:

Step1: Complete the square for x - terms

For the $x^{2}-16x$ part, take half of the coefficient of $x$ (-16), square it. Half of - 16 is -8, and $(-8)^{2}=64$.
$x^{2}-16x=(x - 8)^{2}-64$

Step2: Complete the square for y - terms

For the $y^{2}+6y$ part, half of the coefficient of $y$ (6) is 3, and $3^{2}=9$. So $y^{2}+6y=(y + 3)^{2}-9$.

Step3: Rewrite the original equation

The original equation $x^{2}+y^{2}-16x + 6y+64 = 0$ can be rewritten as:
\((x - 8)^{2}-64+(y + 3)^{2}-9+64=0\)

Step4: Simplify the equation

Combine like - terms:
\((x - 8)^{2}+(y + 3)^{2}-9=0\)
Then \((x - 8)^{2}+(y + 3)^{2}=9\)

Answer:

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