Findng the Nature of Roots

Learn how to use the discriminant to find the nature of roots.

It is easier to know the nature of roots without drawing a graph.

In your grade 11 maths syllabus, you are asked to find the nature of roots of some quadratic expression. NB: The following notes apply only for a quadratic expression.

To solve this question, you need to make use of a discriminant, ∆.

$${\Delta = b^2 -4ac}$$

Before you proceed to answer the question, make sure your expression is written in the form, \(ax^2+bx+c\). Otherwise, do not use the discriminant. If your expression is a quadratic, then do the following:

1. Find the values of \(a\), \(b\), and \(c\) of the quadratic and calculate the value of \(\Delta\).
2. Now that you found the value of \(\Delta\). Note the following:
   1. If \(\Delta\) is a negative, the roots are non-real(imaginary).
   2. Otherwise, the roots are real(\(\Delta \geq 0\)).
3. Now that the roots are real, look at their equality.
   1. If the \(\Delta\) is zero, the roots are equal.
   2. Otherwise, the roots are unequal (\(\Delta > 0\)).
4. If they are unequal, check further if they are rational or irrational.
   1. If the \(\Delta\) is a perfect square, then the roots are rational.
   2. If not a perfect square, then the roots are irrational.

Additionally, pay attention to words like show that. When you are asked this, you must know that all coefficients of the quadratic expression will be given. You then need to substitute the values into your \(\Delta\) equation and find the nature of roots.

The question can also ask you to find values of some unknown parameter, e.g., some \(k\) or whatever parameter. To answer this, know the conditions listed above. In such a question, you will then need to proceed to solve the unknown parameter.