Question

the equation of a circle in expanded form is shown. what is the equation of the circle in star form? use the drop - down menus to complete the equation. $x^{2}+y^{2}-8x + 4y-16 = 0$ click the arrows to choose an answer from each menu. $(\text{choose...})^{2}+(\text{choose...})^{2}=\text{choose...}^{2}$

Explanation:

Step1: Group x and y terms

Rearrange the equation: \(x^{2}-8x + y^{2}+4y=16\)

Step2: Complete the square for x

For \(x^{2}-8x\), take half of -8 (\(-4\)), square it (\(16\)): \(x^{2}-8x + 16=(x - 4)^{2}\)

Step3: Complete the square for y

For \(y^{2}+4y\), take half of 4 (\(2\)), square it (\(4\)): \(y^{2}+4y + 4=(y + 2)^{2}\)

Step4: Add constants to both sides

Add 16 and 4 to the right side: \(16+16 + 4=36\)
So the equation becomes \((x - 4)^{2}+(y + 2)^{2}=36\), and \(36 = 6^{2}\)

Answer:

\((x - 4)^{2}+(y + 2)^{2}=6^{2}\)