Where does the quadratic formula come from?

The quadratic formula is used to find the roots of a quadratic equation which comes in this form:

y = ax² + bx + c

By finding the roots, we mean that we find the values of x when y equals zero. Visually, this is when the graph of the equation crosses the x axis. To do this, we set y = 0 and do a method called 'completing the square'. Then we do a bit of rearranging to make x the subject of the equation, mean x = a number.

Here is the derivation:

y = ax² + bx + c
0 = a(x² + (b/a)x) + c
0 = a((x + (b/2a))² - (b/2a)²) + c
0 = a(x + (b/2a))² - (b²/4a) + c
(b²/4a) - c = a(x + (b/2a))²
b² - 4ac = 4a²(x + (b/2a))²
(b² - 4ac)/4a² = (x + (b/2a))²
((b² - 4ac)/4a² )^1/2 = x + (b/2a)
(b² - 4ac)^1/2/2a = x + (b/2a)
-b/2a + (b² - 4ac)^1/2/2a = x
x = (-b +- (b² - 4ac)^1/2)/2a

This is the quadratic formula! All we do now is substitute in values for a,b and c to get 2 values for x.