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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.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

    Saying D > 0 means no real roots is wrong because the sign rule works the other way round.

Definition

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.

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.

Rule to remember

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

Memory hook

Positive, zero, negative: distinct, equal, none.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Saying D > 0 means no real roots is wrong because the sign rule works the other way round.

Exam tip

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

Quick check

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

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

Answer writing and exam use

1-mark answer

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-mark answer

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

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. D > 0 gives two distinct real roots, D = 0 gives equal real roots, and D < 0 gives no real roots. For 2x^2 - 3x - 2 = 0, we get D = (-3)^2 - 4(2)(-2) = 9 + 16 = 25. Since D > 0, the equation has two distinct real roots. CBSE may ask for the nature of roots directly after giving a discriminant value or after a full equation. Saying D > 0 means no real roots is wrong because the sign rule works the other way round.
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