Nature of Roots  Theory
For the quadratic equation \(a{x^2} + bx + c = 0\), the quadratic formula tells us that the roots are:
\[x = \frac{{  b \pm \sqrt {{b^2}  4ac} }}{{2a}}\]
The term under the squareroot sign, \({b^2}  4ac\), is called the discriminant of the quadratic equation, and is denoted by \(D\). Thus, \(D = {b^2}  4ac\). The discriminant helps us in deciding the nature of the roots.
If \(D > 0\), then there are two real and distinct roots:
\[\alpha = \frac{{  b + \sqrt D }}{{2a}},\,\,\,\beta = \frac{{  b  \sqrt D }}{{2a}}\]
If \(D = 0\), then the two roots turn out to be equal:
\[\alpha = \frac{{  b}}{{2a}},\,\,\,\beta = \frac{{  b}}{{2a}}\]
If \(D < 0\), we have a negative term under the squareroot sign. The roots in this case will exist, but will be nonreal (they will be complex numbers). For example, consider the quadratic equation \({x^2} + x + 1 = 0\). The discriminant in this case is \(D = {1^2}  4 \times 1 \times 1 =  3\), which is negative. The roots are:
\[\alpha = \frac{{  1 + \sqrt {  3} }}{2},\,\,\,\beta = \frac{{  1  \sqrt {  3} }}{2}\]
The roots do exist, but they are not real numbers. You will understand the significance of such quantities when you study complex numbers.
To summarize:

If \(D > 0\), the roots are real and distinct.

If \(D = 0\), the roots are real and equal.

If \(D < 0\), the roots exist but are complex numbers.
Notes:

At this stage, we only deal with quadratics with real numbers as variables and coefficients. However, a quadratic can also have complex numbers as variables and coefficients. You will encounter such equations in a higher class.

If you ever read the statement that the roots of a quadratic equation do not exist, you should immediately realize that this is incorrect. A quadratic equation will always have two roots, and exactly two roots. If D is 0, those two roots happen to be identical. If D is less than 0, real roots do not exist, but the equation still has exactly two roots – which happen to be nonreal complex numbers.