Completing the Square and Finding the Vertex

Perfect Square A polynomial that can be factored into the following form: (x + a) 2 Examples:

Completing the Square x 2 + bx + c is a perfect square if: (The value of c will always be positive.) Ex: Prove the following is a perfect square Half of b=-16 squared is 64=c

Completing the Square Find the c that completes the square: 1.x x + c 2.x 2 – 22x + c 3.x x + c

Factoring a Completed Square If x 2 + bx + c is a perfect square, then it will easily factor to: Ex: Prove the following is a perfect square. Half of b=+8 is +4

Perfect Squares: Parabolas & Circles Find the vertices of the following graphs and state whether they are maximums or minimums. 1.y = (x + 5) 2 – 5 2.y = -(x + 3) y = -3(x – 7) y = 4(x – 52) 2 – 74 State the length of the radius and the coordinates of the center for each circle below: 1.( x – 2 ) 2 + ( y + 7 ) 2 = 64 2.x 2 + y 2 = 36 3.( x + 4 ) 2 + ( y + 11 ) 2 = 5 4.( x + 3 ) 2 + y 2 = 175

y = (x 2 + 8x + ) + 25 – Standard to Graphing: Quadratic y = x 2 + 8x + 25 y = (x 2 + 8x ) y = (x + 4) 2 Vertex: (-4, 9) + 9 Find the vertex of the following equation by completing the square: y = a ( x – h ) 2 + k GOAL Complete the Square: Find the “c” that completes the square Factor what is in the Parentheses Plus a box, minus a box Simplify

Standard to Graphing: Quadratic y = 3x 2 – 18x – 10 y = 3(x 2 – 6x + ) – y = (x – 3) 2 Vertex: (3,-37) 3 – 10 – 27 y = 3(x – 3) 2 – 37 – 3 Find the vertex of the following equation by completing the square: y = a ( x – h ) 2 + k GOAL

A new Equation? What will the graph of the following look like:

Standard to Graphing: Circle x 2 + y 2 + 6x – 12y – 9 = 0 x 2 + 6x + y 2 – 12y – 9 = x 2 + 6x + y 2 – 12y = 9 (x 2 + 6x + ) + (y 2 – 12y + ) = (x + 3) 2 Center: Radius: (-3, 6) + (y – 6) 2 = 54 Find the center and radius of the equation by completing the square: Isolate the terms with variables Arrange similar variables together Complete the square twice