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Quadratic Formula Derivation

The standard form of a quadratic equation is:

ax2+bx+c=0

Here, a,b,c, and a0

We begin by first dividing both sides by the coefficient a

x2+bax+ca=0

We can rearrange the equation:

x2+bax=ca

We can then use the method of completing the square. (Maths is Fun has a really good explanation for this technique)

x2+bax+(b2a)2=ca+(b2a)2

On our LHS, we can clearly recognize that it is the expanded form of (x+d)2 i.e x2+2x·d+d2

(x+b2a)2=ca+b24a2=4ac+b24a2

Taking the square root of both sides

x+b2a=4ac+b22ax=±4ac+b2b2a=b±b24ac2a

This gives you the world famous quadratic formula:

x=b±b24ac2a
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