Nature of Roots - Theory

Nature of Roots - Theory

Go back to  'Quadratic Equations'

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.

 

  
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
More Important Topics
Numbers
Algebra
Geometry
Measurement
Money
Data
Trigonometry
Calculus
Learn from the best math teachers and top your exams

  • Live one on one classroom and doubt clearing
  • Practice worksheets in and after class for conceptual clarity
  • Personalized curriculum to keep up with school

0