Question

How do you derive the quadratic formula?

Answer 1

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!