Question

write the equation in standard form for the circle $x^2 + y^2 + 8x - 33 = 0,$

Explanation:

Step1: Group x-terms and move constant

Group the \(x\)-terms and move the constant to the right: \(x^{2}+8x + y^{2}=33\).

Step2: Complete the square for x

For \(x^{2}+8x\), take half of 8 (\(4\)), square it (\(16\)), add to both sides: \(x^{2}+8x + 16+y^{2}=33 + 16\).

Step3: Factor and simplify

Factor \(x^{2}+8x + 16\) as \((x + 4)^{2}\), so \((x + 4)^{2}+y^{2}=49\).

Answer:

\((x + 4)^{2}+y^{2}=49\)