medium importancemedium difficulty5 min

Nature of Roots Based on Discriminant

The nature of roots depends on the discriminant D: if D > 0 the roots are real and distinct, if D = 0 they are real and equal, and if D < 0 they are not real.

In one line: Positive, zero, negative: distinct, equal, none.

Practice This Concept

Understand it

This rule saves time in exams because the nature of roots can be predicted without solving the whole equation. It is one of the fastest checks in quadratic equations.

Example

If D = 25, the roots are real and distinct; if D = 0, the roots are equal; if D = -9, there are no real roots.

Formula and rule

D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives no real roots.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1For 2x² - 3x - 2 = 0, we get D = (-3)² - 4(2)(-2) = 9 + 16 = 25.
2Since D > 0, the equation has two distinct real roots.

Method you can reuse

1Find D.
2Compare D with zero.
3Use the sign rule to state the nature of roots.
4Mention the exact wording: distinct, equal, or not real.

Visual tip

A three-line sign chart is useful: D > 0, D = 0, D < 0 with the root type written beside each line.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What does D = -9 tell you about the roots of a quadratic equation?

Think first, then show answer

It tells us that the equation has no real roots. A negative discriminant means the roots are not real numbers.

Common wrong answer

Saying D > 0 means no real roots.

Why it loses marks: The sign rule works the other way round.

Don't mix this up

Students sometimes say any positive D gives equal roots, which is incorrect.

Exam tip

Memorise the sign rule carefully: positive means distinct, zero means equal, negative means no real roots.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

CBSE may ask for the nature of roots directly after giving a discriminant value or after a full equation.

1
Enough for 1 mark
The nature of roots depends on the discriminant D: if D > 0 the roots are real and distinct, if D = 0 they are real and equal, and if D < 0 they are not real.
2
Add for 2 marks
Example: If D = 25, the roots are real and distinct; if D = 0, the roots are equal; if D = -9, there are no real roots.
3
Add for 3 marks
It is one of the fastest checks in quadratic equations.

MCQ Quiz

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