How do you derive the quadratic formula?

1 Answer
Jan 10, 2015

This is a little bit tricky but also incredibly elegant!
You start from your general quadratic:
#ax^2+bx+c=0#
Take #c# to the right side:
#ax^2+bx=-c#
The idea is now to transform the left side in something like #(a+b)^2#;
Multiply by #a#;
#a^2x^2+abx=-ac#
Multiply by #4#;
#4a^2x^2+4abx=-4ac#
Add and subtract #b^2# to the left side:
#4a^2x^2+4abx+b^2-b^2=-4ac#
Take the #-b^2# to the right:
#4a^2x^2+4abx+b^2=b^2-4ac#
The left side can be written as:
#(2ax+b)^2=b^2-4ac#
And:
#2ax+b=+-sqrt(b^2-4ac)#
And finally:
#x=(-b+-sqrt(b^2-4ac))/(2a)#
Poetry in algebra!