Nature of Roots - Theory

Go back to  'Quadratic Expressions'

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 square-root 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 square-root sign. The roots in this case will exist, but will be non-real (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:

  1. 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.

  2. 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 non-real complex numbers.

 

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