The roots of a quadratic equation ax^2 + bx + c = 0 are given by x1 = (-b + sqrt (b^2 - 4ac))/2a and x2 = (-b + sqrt(b^2 - 4ac))/2a.
If b^2 - 4ac is negative, we have the square root of a negative number which is an imaginary number.
Complex roots of a quadratic equation are always found in complex-conjugate pairs a + ib and a - ib. The reason behind this is that the coefficients a, b and c in a quadratic equation are real numbers.
Now ax^2 + bx + c = (x - x1)(x - x2) = x^2 - (x1 + x2)x + x1*x2. The term x1 + x2 is real only if x1 and x2 are complex conjugate as the imaginary part of the complex numbers is eliminated. Similarly, x1*x2 gives a real result if the two are complex conjugate pairs.
Complex roots of a quadratic equation are found in pairs complex conjugate numbers because the coefficients of a quadratic equation are real numbers.