We'll start from the quadratic equation.
ax^2 + bx + c = 0
We'll factorize by a:
a(x^2 + bx/a + c/a) = 0
We'll group the first 2 terms and we'll complete the square:
a(x^2 + 2*x*b/2a + b^2/4a)- b^2/4a + c/a = 0
The first 3 terms represent the perfect square (x +b/2a)^2:
a(x +b/2a)^2 = b^2/4a - c/a
a(x +b/2a)^2 = (b^2 - 4c)/4a
We'll divide by a:
(x +b/2a)^2 = (b^2 - 4c)/4a^2
We'll apply square root both sides:
x + b/2a = +/-sqrt(b^2 - 4c)/2a
We'll subtract b/2a:
x = [-b +/- sqrt(b^2 - 4c)]/2a
Note: the square root exists if the expression under the square root is positive or zero.
the expression under the square root is delta, the discriminant of equation.
If delta > 0=> the equation has 2 distinct roots.
If delta = 0 => the eq. has 2 equal roots.
If delta < 0 => the eq. has no ral roots.