5.8 Complete the Square This is the second to last lesson for MT5. Our last lesson will be on solving word problems. This lesson is on “Completing the Square” and is used to find the Vertex Form of a quadratic function. This lesson will show you how to turn this… x 2 + 6x – 16 = 0 Into this… y = (x + 3) The vertex form, y = a(x – h) 2 + k, is used as a shortcut for graphing parabolas (and we all like short cuts!)

5.8 Complete the Square Let’s start with our example problem and make a plan to eventually get something that looks like this (x – h) 2. This is called a perfect square. Let’s begin Start with: x 2 + 6x – 16 = 0 Step 1: move “C” (is this case, -16) to the other side of the “=“ sign. Use inverse property x 2 + 6x = 16 Notice the -16 becomes +16! Step 2: Leave some space and put “_____” in place where you need to fill in numbers. We are going to create a “perfect square” so create a place for that too. + ___ + ___ (x + ___) 2 = ___ You get… Be sure to copy this setup. You will use it for every question.

5.8 Complete the Square The setup will be the same every time. Please memorize this setup. x 2 + 6x = 16 Step 1: Move “C” Step 2: Create space with “___” and setup for perfect square + ___ + ___ (x + ___) 2 = ___ Now Step 3: Fill in your numbers x 2 + 6x = 16 + ___ + ___ (x + ___) 2 = ___ Find this one first! Take this number and divide by 2 6/2=3 3

5.8 Complete the Square Let’s do it again without the arrows… x 2 + 6x = 16 + ___ + ___ (x + ___) 2 = ___ 6 = Now fill in the rest… 3 2 Is = Now it almost looks like vertex form y = a(x – h) 2 + k

5.8 Complete the Square Last thing to do is move the 25 back and add a y =… x 2 + 6x = 16 + ___ + ___ (x + ___) 2 = ___ y = (x + 3) Notice there is no “a” when a = 1 Remember, the vertex form, y = a(x – h) 2 + k,

5.8 Complete the Square Here’s another problem from beginning to end… Ex 2: x 2 – 8x + 12 = 0 x 2 – 8x + ___ = ___ (x - ___) 2 = ___8/2 = y = (x – 4) 2 – 4

5.8 Complete the Square And another one… Ex 3: x 2 – 12x + 32 = 0 x 2 – 12x + ___ = ___ (x - ___) 2 = ___12/2 = y = (x – 6) 2 – 4