How are used imaginary numbers in solving quadratic equations ?

It is vague to say imaginary numbers are used in quadratic
equations. A quadratic equation is 2nd degree polynomial equation , ax^2+bx+c = 0 ,where
the quadratic coefficient of x^2  a , the linear coefficient x , b and  the constant
term c are all real.

But in quadratic equations  the
solutions or the roots could be imaginary or complex.

 We
know  ax^2+bx+c = 0 is a general quadratic equation. Then a(x^2+bx/a)+c/a =
0

x^2+bx/a +((b/2a)^2 -(b/2a)^2 + c/a =
0

(x+b/2a)^2  =
(b^2-4ac)/(2a)^2

 x +b/2a =
+(1/2a)sqrt(b^2-4ac).

x+b/2a =
-(1/2a)sqrt(b^2-4ac).

Or

x =
-(b/2a) + (1/2a)(b^2-4ac)..........(1) Or

x = -(b/2a) -
(1/2a) (b^2-4ac)..........(2)

From (1) and (2) it is only
when b^2-4ac is negative that an imaginary roots can
occur.

Also it could be noticed that the imaginary roots
can occur in conjugate pairs  like  A+Bi and A-Bi and never in single imagiginary root.
The idea is usuful while solving the quadratic equation.