1. Complete the Square – Conics
I.. Completing the square of a parabola.
Also called the Vertex form of a parabola.
A) Turns y = ax2 + bx + c into y = a(x – h)2 + k. Or
x = ay2 + by + c into x = a(y – k)2 + h.
1) Converts standard form into vertex form.
2) Quickly finds the vertex point (given the vertex form).
a) Vertex is (h , k) for y = a(x – h)2 + k
b) Vertex is (h , k) for x = a(y – k)2 + h
1. Remember to change the sign of h.

2. Complete the Square – Conics
II.. Completing the square for Parabolas.
(½ the number, [“a” times] square the number, move it on back)
A) Rewrite the equation as … everything else = non-squared term.
1) Make the coefficient of the non-squared term = 1.
You now have ax2 + bx + c = y or ay2 + by + c = x.
B) Group the same variables together and factor out the “a” term.
1) a(x2 + b/ax) + c = y or a(y2 + b/ay) + c = x (b/a is the new “b” term)
C) Take ½ of the new “b” term.
1) write it in a(x + #)2 + c = y form or a(y + #)2 + c = x form.
D) Square the number, but multiply it by the “a” term and write it on the other side of the = sign. a(x + #)2 + c = y + a(#2)
E) Move the # you just wrote to the other side of the = sign
and change its sign (always makes it a negative). Collect like terms.

3. Complete the Square – Conics
Example: x2 – y + 12x + 7 = 0
A) Change it to x2 = y x2 + 12x + 7 = y
B) Factor out the “a” term: (x2 + 6x) + 7 = y
C) Take ½ the # (x + 3)2 + 7 = y
D) Square the # (times “a”) 2(x + 3)2 + 7 = y + 2(32)
E) Move it on back –18
and collect like terms (x + 3)2 – 11 = y

4. Complete the Square – Conics
Example: y2 – 2x – 24y + 34 = 0
A) Change it to y2 = x y2 – 24y + 34 = 2x
And make it say 1x (÷ by 2) 3y2 – 12y + 17 = x
B) Factor out the “a” term: (y2 – 4y) + 17 = x
C) Take ½ the # (y – 2) = x
D) Square the # (times “a”) 3(y – 2) = x + 3(-22)
E) Move it on back –12
and collect like terms (y – 2)2 + 5 = x

5. Complete the Square – Conics
III.. Completing the square for Circles, Ellipses and Hyperbolas.
(½ the number, [“a” times] square the number, get in standard form)
A) Rewrite the equation so it is in … ax2 & by2 terms = # … form.
ax2 + by2 + cx + dy = #
B) Group the x’s together and the y’s together in parenthesis.
(ax2 + cx) + (by2 + dy) = #
C) Factor out the terms in front of the x2 and y2 from their groups.
Ex: (3x2 + 12x) + (4y2 – 24y) =  3(x2 + 4x) + 4(y2 – 6y) = 7
D) Take ½ the number and write it in a(x + #)2 + a(x + #)2 = # form.
Ex: 3(x + 2)2 + 4(y – 3)2 = 7
E) Square the number, but multiply it by the “a” term and write it on the other side of the = sign. Ex: 3(x + 2)2 + 4(y – 3)2 = 7 + 3(22) + 4(-32)
F) Collect like terms and write it in standard form: Circle: (x – h)2 + (y – k)2 = r2
Ellipse: (x – h)2 + (y – k)2 = Hyperbola: (x – h)2 – (y – k)2 = 1 or – (x – h)2 + (y – k)2 = 1
# # # # # #

6. Complete the Square – Conics
Example (circle): x2 + 4y2 + 16x – 24y – 48 = 0
A) Change it to x2 + 4y2 + 16x – 24y = 48
B) Group them (4x2 + 16x) + (4y2 – 24y) = 48
C) Factor out 1st # (x2 + 4x) + 4(y2 – 6y) = 48
D) ½ the number (x + 2)2 + 4(y – 3)2 = 48
E) Sq the # (times “a”) 4(x + 2)2 + 4(y – 3)2 = (22) + 4(-32)
F) Collect like terms (x + 2)2 + 4(y – 3)2 = 100
& get in standard form 4(x + 2)2 + 4(y – 3)2 = 100
(x + 2)2 + (y – 3)2 = 25

7. Complete the Square – Conics
Example (ellipse): x2 + 6y2 + 24x – 24y + 12 = 0
A) Change it to x2 + 6y2 + 24x – 24y = –12
B) Group them (4x2 + 24x) + (6y2 – 24y) = –12
C) Factor out 1st # (x2 + 6x) + 6(y2 – 4y) = –12
D) ½ the number (x + 3)2 + 6(y – 2)2 = –12
E) Sq the # (times “a”) 4(x + 3)2 + 6(y – 2)2 = –12 + 4(32) + 6(-22)
F) Collect like terms (x + 3)2 + 6(y – 2)2 = 48
& get in standard form (x + 3)2 + 6(y – 2)2 = 48

8. Complete the Square – Conics
Example (hyperbola): 4x2 – 6y2 + 24x – 24y – 12 = 0
A) Change it to x2 – 6y2 + 24x – 24y = 12
B) Group them (4x2 + 24x) + (–6y2 – 24y) = 12
C) Factor out 1st # (x2 + 6x) – 6(y2 + 4y) = (watch the signs here)
D) ½ the number (x + 3)2 – 6(y + 2)2 = 12
E) Sq the # (times “a”) 4(x + 3)2 – 6(y + 2)2 = (32) + –6(22)
F) Collect like terms (x + 3)2 – 6(y + 2)2 = 24
& get in standard form 4(x + 3)2 – 6(y + 2)2 = 24