Question

the equation of a circle in expanded form is shown. what is the equation of the circle in standard 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, square it: \((\frac{-8}{2})^{2}=16\). Add 16 to both sides: \(x^{2}-8x + 16+y^{2}+4y=16 + 16\)

Step3: Complete the square for y

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

Step4: Write as perfect squares

\((x - 4)^{2}+(y + 2)^{2}=36\) (since \(36 = 6^{2}\))

Answer:

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