Complete the Square Conics I Completing the square

Complete the Square – Conics I. . Completing the square of a parabola. Also called the Vertex form of a parabola. A) Turns y = ax 2 + bx + c into y = a(x – h)2 + k. Or x = ay 2 + 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.

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 ax 2 + bx + c = y or ay 2 + by + c = x. B) Group the same variables together and factor out the “a” term. 1) a(x 2 + b/ax) + c = y or a(y 2 + 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.

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

Complete the Square – Conics Example: 6 y 2 – 2 x – 24 y + 34 = 0 A) Change it to y 2 = x 6 y 2 – 24 y + 34 = 2 x And make it say 1 x (÷ by 2) 3 y 2 – 12 y + 17 = x B) Factor out the “a” term: 3(y 2 – 4 y) + 17 = x C) Take ½ the # 3(y – 2)2 + 17 = x D) Square the # (times “a”) 3(y – 2)2 + 17 = x + 3(-22) E) Move it on back and collect like terms – 12 3(y – 2)2 + 5 = x

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 … ax 2 & by 2 terms = # … form. ax 2 + by 2 + cx + dy = # B) Group the x’s together and the y’s together in parenthesis. (ax 2 + cx) + (by 2 + dy) = # C) Factor out the terms in front of the x 2 and y 2 from their groups. Ex: (3 x 2 + 12 x) + (4 y 2 – 24 y) = 7 3(x 2 + 4 x) + 4(y 2 – 6 y) = 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: Ellipse: (x – h)2 + (y – k)2 = 1 # # (x – h)2 + (y – k)2 = r 2 Hyperbola: (x – h)2 – (y – k)2 = 1 or – (x – h)2 + (y – k)2 = 1 # #

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

Complete the Square – Conics Example (ellipse): 4 x 2 + 6 y 2 + 24 x – 24 y + 12 = 0 A) Change it to 4 x 2 + 6 y 2 + 24 x – 24 y = – 12 B) Group them (4 x 2 + 24 x) + (6 y 2 – 24 y) = – 12 C) Factor out 1 st # 4(x 2 + 6 x) + 6(y 2 – 4 y) = – 12 D) ½ the number 4(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 4(x + 3)2 + 6(y – 2)2 = 48 & get in standard form 4(x + 3)2 + 6(y – 2)2 = 48 48

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