How are completing the square and quadratic formula related?

Question

3915 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-06-10T23:10:14

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:

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