How are completing the square and quadratic formula related?

1 Answer
Jun 10, 2015

Answer:

The quadratic formula can be deduced from generically completing the square.

Explanation:

Given a generic quadratic equation:

#f(x) = ax^2 + bx + c = 0#

Notice that:

#a(x+b/(2a))^2 = a(x^2 + 2b/(2a)x + b^2/(2a)^2)#

#=ax^2+bx + b^2/(4a)#

So

#f(x) = ax^2 + bx + c = a(x+b/(2a))^2 + (c-b^2/(4a))#

If #f(x) = 0# then

#a(x+b/(2a))^2 + (c-b^2/(4a)) = 0#

Add #(b^2/(4a) - c)# to both sides to get:

#a(x+b/(2a))^2 = (b^2/(4a) - c) =(b^2-4ac)/(4a)#

Divide both sides by #a# to get:

#(x+b/(2a))^2 = (b^2-4ac)/(4a^2) = (b^2-4ac)/((2a)^2)#

Hence:

#x + b/(2a) = +- sqrt((b^2-4ac)/((2a)^2)) = +-sqrt(b^2-4ac)/(2a)#

Subtract #b/(2a)# from both sides to get:

#x = (-b+-sqrt(b^2-4ac))/(2a)#