1. Module 5 Topic D

2. Learning Target
I will be able to: Write the equations of circles and graph circles.
I will be able to: Use the equation and graph of a circle to solve problems.

8. Learning Target
I will be able to: Complete the square in order to write the equation of a circle in center-radius form.
I will be able to: recognize when a quadratic in x and y is the equation of a circle.

11. Example 1
The following is the equation of a circle with radius 5 and center 1, 2 . Do you see why?
𝑥 2 −2𝑥+1+ 𝑦 2 −4𝑦+4=25
We know that the equation 𝑥 2 −2𝑥+1+ 𝑦 2 −4𝑦+4=25 is a circle with radius 5 and center (1, 2) because when we multiply out the equation 𝑥− 𝑦−2 2 = 5 2 , we get 𝑥 2 −2𝑥+1+ 𝑦 2 −4𝑦+4=25

15. M5TD - Complete the Square
Rewrite the quadratic so only the first two terms are on the left of the equal sign.
Make sure the leading coefficient is 1 or else divide everything by the leading coefficient to make it that way
Add ½ of the second coefficient all squared to both sides.
Rewrite the left hand side as a square
Square root both sides
Solve
x2 + 6x = 16
It already is
x2 + 6x =
x2 + 6x + 32 =
(x + 3)2 = 25
x + 3 =  5
x = -3  5
x = 2, -8

16. Supplemental Information
M5TD
Supplemental Information

22. M5TD - Complete the Square
The Perfect Square
(x + 3)2
(x + k)2 = x2 + 2kx + k2
=(x + 3)(x + 3)
= x2 + 2(3x) + 32
= x2 + 6x + 9

23. M5TD - Complete the Square
What should c be to make a perfect square
x2 + 8x + c
you have to add (8/2)2 = 42 = 16 to get a perfect square or (x + 4)2

24. M5TD - Complete the Square
Rewrite the quadratic so only the first two terms are on the left of the equal sign.
Make sure the leading coefficient is 1 or else divide everything by the leading coefficient to make it that way
Add ½ of the second coefficient all squared to both sides.
Rewrite the left hand side as a square
Square root both sides
Solve
x2 + 6x = 16
It already is
x2 + 6x =
x2 + 6x + 32 =
(x + 3)2 = 25
x + 3 =  5
x = -3  5
x = 2, -8