Be The Professor- Completing the Square

Completing the Square 

First, the goal of the mathematical concept, completing the square is to take a quadratic function, and put it in vertex form.  

Steps to take when completing the square using an example:  

x² + 16x+ 2

1) divide the middle term by two to get its half.

16/2 = 8 

2) Create a zero in the function.  This is done by taking the value found in step 1 and squaring, and then adding and subtracting your new number to the equation.

8²=64

x²+16x+64-64+2

3) Find a perfect square factor

Perfect square factor: (x²+16x+64) à (x+8)²

4) Write the problem in vertex form.

Vertex Form= f(x) = (x+8)² – 62

Vertex:  (-8, -62)

Example using a quadratic function with an "a" value of more than 1 

f(x) = 2x²+4x-5

Factor out the leading coefficient, in this case the 2, from the equation

2(x²+2x)-5

Take the half of the middle value, now a 2 instead of a 4 thanks to factoring.

2/2=1

create a zero:  f(x)= 2(x²+2x+1-1)-5

Perfect square factor: 2(x+1)² – 1(2)-5

Vertex Form: 2(x+1)² – 7

Vertex: (-1, -7)

Now that you've completed the square, you can go on to graph the function using the vertex you found!