Mathematics is a unique subject within which many equations that you encounter have lots of functions and can be derived from the basics. The quadratic formula for instance is incredibly useful to quickly solve quadratics (with real roots) that cannot be factorised: It seems at first very tricky indeed at the first glance on how to start, however it only requires GCSE knowledge for the proof; completing the square. We start with ax2+bx+c=0, where y=0 and the next steps follow: 1. Factorise out the 'a' a(x2+bx/a) +c=0. 2. Half the inside number&square the bracket&subtract square from inside to make the function equal to the previous line a[(x+b/2a)2-(b/2a)2] +c=0. 3. Expand the brackets a(x+b/2a)2-b2/4a +c=0. 4. Keep the 'x' term to one side a(x+b/2a)2. =b2/4a - c. 5. Combine the fractions a(x+b/2a)2 =(b2-4ac) /4a. 6. Divide by 'a' (x+b/2a)2 =(b2-4ac) /4a2 7. Square root both sides x+b/2a = √(b2-4ac) / 2a. 8. Make 'x' on its own by subtracting b/2a from both sides. Voila! x = [-b± √(b2-4ac)] / 2a.