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Completing the Square and Finding the Vertex

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.

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Completing the Square and Finding the Vertex

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  1. Completing the Square and Finding the Vertex

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

  3. Completing the Square x2 + 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

  4. Completing the Square Find the c that completes the square: • x2 + 50x + c • x2 – 22x + c • x2 + 15x + c

  5. Factoring a Completed Square If x2 + 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

  6. Perfect Squares: Parabolas & Circles Find the vertices of the following graphs and state whether they are maximums or minimums. • y = (x + 5)2 – 5 • y = -(x + 3)2 + 1 • y = -3(x – 7)2 + 8 • y = 4(x – 52)2 – 74 State the length of the radius and the coordinates of the center for each circle below: • ( x – 2 )2 + ( y + 7 )2 = 64 • x2 + y2 = 36 • ( x + 4 )2 + ( y + 11 )2 = 5 • ( x + 3 )2 + y2 = 175

  7. Standard to Graphing: Quadratic Find the vertex of the following equation by completing the square: y = x2 + 8x + 25 Find the “c” that completes the square GOAL y = a ( x – h )2 + k Plus a box, minus a box Complete the Square: y = (x2 + 8x ) + 25 y = (x2 + 8x + ) + 25 – 16 16 Factor what is in the Parentheses y = (x + 4)2 + 9 Simplify (-4, 9) Vertex:

  8. Standard to Graphing: Quadratic Find the vertex of the following equation by completing the square: y = 3x2 – 18x – 10 GOAL y = a ( x – h )2 + k y = 3(x2 – 6x + ) – 10 – 3 9 9 y = (x – 3)2 3 – 10 – 27 y = 3(x – 3)2 – 37 (3,-37) Vertex:

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

  10. Standard to Graphing: Circle Find the center and radius of the equation by completing the square: x2 + y2 + 6x – 12y – 9 = 0 Arrange similar variables together x2 + 6x + y2 – 12y – 9 = 0 Isolate the terms with variables + 9 + 9 x2 + 6x + y2 – 12y = 9 Complete the square twice (x2 + 6x + ) + (y2 – 12y + ) = 9 + + 9 36 9 36 (x + 3)2 + (y – 6)2 = 54 (-3, 6) Center: Radius:

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