JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Nature of Roots

In a quadratic equation \[a{{x}^{2}}+bx+c=0\], let us suppose that \[a,b,c\] are real and \[a\ne 0\]. The following is true about the nature of its roots.

(1) The equation has real and distinct roots if and only if \[D\equiv {{b}^{2}}-4ac>0\].

(2) The equation has real and coincident (equal) roots if and only if \[D\equiv {{b}^{2}}-4ac=0\].

(3) The equation has complex roots of the form \[\alpha \pm i\beta ,\,\alpha \ne 0,\] \[\beta \ne 0\in R\]  if and only if \[D\equiv {{b}^{2}}-4ac<0.\]

(4) The equation has rational roots if and only if \[a,b,c\in Q\] (the set of rational numbers) and \[D\equiv {{b}^{2}}-4ac\] is a perfect square (of a rational number).

(5) The equation has (unequal) irrational (surd form) roots if and only if \[D\equiv {{b}^{2}}-4ac>0\] and not a perfect square even if a, b and c are rational. In this case if \[p+\sqrt{q}\], \[p,q\] rational is an irrational root, then \[p-\sqrt{q}\] is also a root (a, b, c being rational).

(6) \[\alpha +i\beta \] (\[\beta \ne 0\] and \[\alpha ,\beta \in R\]) is a root if and only if its conjugate \[\alpha -i\beta \] is a root, that is complex roots occur in pairs in a quadratic equation. In case the equation is satisfied by more than two complex numbers, then it reduces to an identity. \[0.{{x}^{2}}+0.x+0=0\], i.e., \[a=0=b=c\].