College Algebra & Trigonometry, 2018a

30 CHAPTER 1. ALGEBRA REVIEW
Then, we will move the c to the other side of the equation to clear out some room
a
for completing the square:
x 2 + b a x + c a =0
− c a = − c a
x 2 + b a x
= − c a
Now we need to complete the square. If you are already familiar with this process,
you may wish to skip the following explanation.
If we look at what happens when we square a binomial like (x+3) 2 , we will begin
to notice a pattern.
(x +3) 2 =(x +3)(x +3)=x 2 +6x +9
(x +4) 2 =(x +4)(x +4)=x 2 +8x +16
(x +5) 2 =(x +5)(x +5)=x 2 +10x +25
(x +6) 2 =(x +6)(x +6)=x 2 +12x +36
Our goal in the derivation of the Quadratic Formula is to rewrite the expression
x 2 + b x as a perfect square in the form (x+
a )2 . The reason that we want to do
this is that writing an expression as a binomial squared eliminates the problem of
having both an x and an x 2 , which was preventing us from getting the x by itself
in the standard quadratic equation.
If we can figure out what should take the place of the blanks in the statement:
x 2 + b x+ = (x+ )2
a
then we will be well on our way to deriving the quadratic formula.