Where does the quadratic equation come from?

  • Google+ icon
  • LinkedIn icon

This is a quick derivation of the quadratic formula for an GCSE Maths students looking to stretch themselves a little and go for that A*!

If we have a general quadratic equation:

ax2+bx+c=0   (eq1)

You will have been told that the solution is:

x= (-b +/- sqrt[b^2-4ac])/2a   (eq 2)

But why is this true? Let's go through this in steps. Firstly, we will divide our equation by a:

x2+(b/a)x+(c/a) =0   (eq 3)

Now we can use a clever trick called 'completing the square', which states that:

x2+(something)x = (x+(something)/2)2 - (something/2)2   (eq 4)

You can check this by multiplying out the brackets on the right hand side!

Anyway, this clever trick tells us that:

x2+(b/a)x+(c/a) = (x+(b/2a))2 - (b/2a)2 + (c/a) = 0   (eq 5)

Let's tidy up by moving the contants over to the right hand side:

(x+(b/2a))2 = (b/2a)2 - (c/a)   (eq 6)

Now we take the sqaure root of both sides:

(x+(b/2a)) = +/- sqrt[(b/2a)2 - (c/a)]   (eq 7)

The reason for the +/- is that there are always two answers to the sqrt (for example, (2)= 4, but so does (-2)2, so sqrt[4] = +/- 2 ). Let's look at the square root on the right hand side and tidy it up a bit:

sqrt[(b/2a)2 - c] = sqrt[b2/4a2 - c] = sqrt[(b2 - 4ac)/4a2]   (eq 8)

Where in the last step I took out a factor of 1/4a2. We can now use that sqrt[AB] = sqrt[A]sqrt[B] to write:

sqrt[(b2 - 4ac)/4a2] = sqrt[1/4a2]sqrt[(b2 - 4ac)]   (eq 9)

But 1/4a2 is just (1/2a)2, so sqrt[1/4a2] = 1/2a !

So we have:

sqrt[1/4a2]sqrt[(b2 - 4ac)] = (1/2a)sqrt[(b2 - 4ac)]   (eq 10)

Phew, we're nearly there now! Using this into equation (7) we have:

(x+(b/2a)) = +/- (1/2a)sqrt[(b2 - 4ac)]   (eq 11)

Let's subtract (b/2a) from both sides to make x the subject:

x = (-b/2a) +/- (1/2a)sqrt[b2 - 4ac]   (eq 12)

And now (last step!) we take out a factor of (1/2a) and we have at last:

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

And we're done!

Ryan M. Mentoring Physics tutor, GCSE Maths tutor, GCSE Science tutor...

About the author

is an online GCSE Maths tutor with MyTutor studying at Cambridge University

Still stuck? Get one-to-one help from a personally interviewed subject specialist.

95% of our customers rate us

Browse tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss