Question

How are completing the square and quadratic formula related?

Answer 1

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)`