How do you derive the quadratic formula?

The quadratic formula is the formula that solves any quadratic equation. So, keeping things general, let's begin with a completely general quadratic equation:

ax2+bx+c=0

This is completely general because any quadratic equation can be rearranged into this form. If you find that you can't get your equation into this form, it probably isn't quadratic!

Ultimately, we want to rearrange to make x the subject. 

First, we divide through by a:

x+ (b/a)x + (c/a) = 0


Now, we 'complete the square':

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

Rearranging and putting the two terms outside the bracket above a common denominator: 

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

Taking the square root of both sides:

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

(where +- indicates we can take either the positive or negative solution)

Finally, rearranging for x:

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

Voila! Apologies for all the brackets! 

x= (-b

Answered by Aneesh N. Maths tutor

4823 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

How do you differentiate 2^x?


1. (a) Express 7cosx - 24sin x in the form R cos (x + a), (b) hence what is the minimum value of this equation


The finite region S is bounded by the y-axis, the x-axis, the line with equation x = ln4 and the curve with equation y = ex + 2e–x , (x is greater than/equal to 0). The region S is rotated through 2pi radians about the x-axis. Use integration to find the


Given that z=sin(x)/cos(x), show that dz/dx = sec^2(x).


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences