What is the quadratic formula and how is it derived?

1 Answer
Oct 25, 2015

For any general quadratic equation of the form #ax^2+bx+c=0#, we have the quadratic formula to find the values of x satisfying the equation and is given by
#x=(-b+-sqrt(b^2-4ac))/(2a)#

To derive this formula, we use completing the square in the general equation #ax^2+bx+c=0#

Dividing throughout by a we get : #x^2+b/ax+c/a=0#

Now take the coefficient of x, half it, square it, and add it to both sides and rearrange to get

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

Now right the left hand side as a perfect square and simplify the right hand side.

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

Now taking the square root on both sides yields :

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

Finally solving for x gives

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

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