# How do you derive the quadratic formula?

##### 1 Answer

Feb 17, 2016

See explanation...

#### Explanation:

Given

Note that:

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

So:

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

Add

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

Divide both sides by

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

Take the square root of both sides (allowing for either sign) to get:

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

Subtract

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

Note that all of this is based on simple properties of arithmetic, so will work regardless of whether