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.
( choose... )$^2 + ($ choose... $)^2 = $ 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). Add 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 (2), square it (4). Add to both sides: \(x^2 - 8x + 16 + y^2 + 4y + 4 = 16 + 16 + 4\)

Step4: Factor perfect squares

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

Answer:

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