Completing the Square

What do you get when you foil the following expressions? (x + 1) (x+1)= (x + 6)2 = (x + 7)2 = (x + 2) (x+2) = (x + 8)2 = (x + 3) (x+3) = (x + 4) (x+4) = (x + 9)2 = (x + 5) (x+5) = (x + 10)2 =

What do you get when you foil the following expressions? x2 + 2x + 1 (x + 10)2 = x2 + 20x + 100 (x + 2)2 = x2 + 4x + 4 (x - 13)2 = x2 - 26x + 169 (x - 3)2 = x2 - 6x + 9 (x - 25)2 = x2 - 50x + 625 (x - 4)2 = x2 - 8x + 16 (x – 0.5)2 = x2 - x + 0.25 x2 – 6.4x + 10.24 (x + 5)2 = x2 + 10x + 25 (x – 3.2)2 =

Fill in the missing number to complete a perfect square. x2 + 2x + ____ x2 - 14x + ___ x2 + 8x + ___ x2 – 20x + ___ x2 + 6x + ___ x2 + 16x + _____

Fill in the missing number to complete a perfect square. x2 + 10x + 25 x2 + 10x + ___ = (x + 5)2 x2 - 30x + 225 x2 - 30x + ___ = (x - 15)2 x2 – 2.8x + 1.96 x2 – 2.8x + ___ = (x – 1.4)2 x2 + 18x + 81 x2 + 18x + ___ = (x + 9)2 x2 + 12x + ___ x2 + 12x + 36 = (x + 6)2 x2 + 0.5x + _____ x2 + 0.5x + 0.0625 = (x – 0.25)2

Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 + 14x - 10 y = x2 + 14x + ____ - 10 y = x2 + 14x + 49 - 10 - 49 y = (x + 7)2 -59 The vertex is at (-7, -59)

Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 - 12x + 5 y = x2 - 12x + ____ + 5 y = x2 - 12x + 36 + 5 - 36 y = (x - 6)2 - 31 The vertex is at (6, -31)

Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 - 28x + 200 The vertex is at (14, 4) y = x2 - 28x + ____ + 200 y = x2 - 28x + 196 + 200 - 196 y = (x - 14)2 + 4

Changing from standard form to vertex form By completing the square on a quadratic in standard form, it is changed into vertex form Change to vertex form: y = x2 – 0.75x - 1 y = x2 – 0.75x + ____ + - 1 y = x2 – 0.75x + .140625 - 1 - .140625 The vertex is at (0.375, -1.140625) y = (x – 0.375)2 – 1.140625

Change to vertex form: y = x2 + 4x + 10 y = x2 + 4x + ___ + 10 y = x2 + 4x + 4 + 10 - 4 y = (x + 2)2 + 6

Change to vertex form: y = x2 + 19x - 1 y = x2 + 19x + ___ - 1 y = x2 + 19x + 90.25 - 1 – 90.25 y = (x + 9.5)2 - 91.25

More Complicated Versions of Completing the Square If the leading coefficient is not equal to 1, completing the square is slightly more difficult. Directions for Completing the Square: 1.) Move the constant out of the way. 2.) Factor out A from the x2 and x term. 3.) Determine what is half of the remaining B. 4.) Square it and put this in for C. 5.) Put in a constant to cancel out the last step. 6.) Write the parenthesis as a perfect square and simplify everything else.

Change to vertex form: y = 2x2 + 4x + 10 Vertex at (-1, 8)

Change to vertex form: y = 3x2 + 12x + 22 Vertex at (-2, 10) y = 3(x + 2)2 + 10

Change to vertex form: y = 6x2 - 48x + 65

Change to vertex form: y = 7x2 - 98x + 400

Change to vertex form: y = 12x2 - 60x + 312

Change to vertex form: y = -5x2 + 20x - 32 Vertex at (2, -12) y = -5(x - 2)2 - 12

Change to vertex form: y = -6x2 + 72x - 53 Vertex at (6, 163) y = -6(x2 - 12x + 36) - 53 + 216 y = -6(x - 6)2 + 163

Methods of Locating the Vertex of a Parabola: If the quadratic is in vertex form: 𝑦=𝑎 𝑥−ℎ 2 +𝑘 The vertex is @ (h, k): If the quadratic is in factored form: The x value of the vertex is halfway between the roots. Plug in & solve to find the y value. 𝑦=𝑎 𝑥−__ 𝑥−__ If the quadratic is in standard form: Complete the square to change to vertex form. 𝑦=𝑎 𝑥 2 +𝑏𝑥+𝑐

Change to vertex form: Vertex at (-0.3, -2.45)

Change to vertex form:

Change to vertex form:

Change to vertex form:

Solve by completing the square.

Solve by completing the square.

Example: Solve by completing the square: x2 + 6x – 8 = 0

Solve by completing the square:

Solve by completing the square:

Solve by completing the square: This is called the Quadratic Formula. You must memorize it!!!