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:


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

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