Derive the quadratic formula. From it, write down the determinant and explain, how is it related to the roots of a quadratic equation.

The general quadratic equation is ax^2+bx+c=0, now we need to Complete The Square, which allows us to have only one "x" term. First, divide the equation by "a", leaving us with x^2... x^2+(b/a)x+c/a=0. Now C.T.S. (x^2+(b/a)x + (b/2a)^2) + c/a - (b/2a)^2 = 0. Move all non-"x" terms to the other side and simplify first bracket (x+(b/2a))^2 = (b/2a)^2 - c/a. Putting right side over the same denominator gives = (b^2 - 4ca)/4a^2. Taking square root of both sides leaves us with x+b/2a = +- sqrt(b^2-4ca)/2a. Then we just move b/2a to the other side and we get the quadratic formula. The determinant is "b^2-4ca". Depending on its value, it can give us 3 different types of roots. If it is positive, we will get 2 real roots. If it is zero, we get one root that is equal to -b/2a. If it is negative, we will get 2 complex roots.