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Discriminant

The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac.

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Student-friendly explanation

The discriminant tells us about the roots before we actually solve the equation. By checking D, we can see whether the roots are real, equal, or not real.

How to write this in exams

  1. 1

    Start with the exact idea

    The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac.

  2. 2

    Then show how to use it

    1. Identify a, b, and c. 2. Substitute them into D = b^2 - 4ac. 3. Calculate carefully with signs. 4. Interpret the result if asked.

  3. 3

    Add one concrete example

    For x^2 - 4x + 1 = 0, D = (-4)^2 - 4(1)(1) = 12.

  4. 4

    Avoid this incomplete answer

    Writing D = 4 - 12 for 3x^2 + 2x + 7 = 0 is wrong because 4ac was not calculated correctly.

Definition

The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac.

Example

For x^2 - 4x + 1 = 0, D = (-4)^2 - 4(1)(1) = 12.

Rule to remember

D = b^2 - 4ac. Use the coefficients from standard form and keep the sign of b exactly as it appears in the equation.

Memory hook

Discriminant decides the kind of roots.

Examples and method

Worked example

For x^2 - 4x + 1 = 0, a = 1, b = -4, c = 1. So D = (-4)^2 - 4(1)(1) = 16 - 4 = 12. Since D is positive, the equation has two real roots.

Method to apply

1. Identify a, b, and c. 2. Substitute them into D = b^2 - 4ac. 3. Calculate carefully with signs. 4. Interpret the result if asked.

Diagram support

A small table for a, b, c, then D, helps reduce sign mistakes in exam work.

How CBSE asks it

Students may be asked to calculate D, interpret its value, or use D to predict the nature of roots.

Avoid common mistakes

Common confusion

Students often forget that b includes its sign and write b^2 with the wrong value.

Common wrong answer

Writing D = 4 - 12 for 3x^2 + 2x + 7 = 0 is wrong because 4ac was not calculated correctly.

Exam tip

Use the discriminant first when the question asks about the nature of roots.

Quick check

What is the discriminant of 3x^2 + 2x + 7 = 0?

D = b^2 - 4ac = 2^2 - 4(3)(7) = 4 - 84 = -80. So the discriminant is -80, and it is negative.

Answer writing and exam use

1-mark answer

The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac.

2-mark answer

The discriminant of a quadratic equation ax^2 + bx + c = 0 is D = b^2 - 4ac. D = b^2 - 4ac. Use the coefficients from standard form and keep the sign of b exactly as it appears in the equation. For x^2 - 4x + 1 = 0, D = (-4)^2 - 4(1)(1) = 12.

3-mark answer

The discriminant tells us about the roots before we actually solve the equation. By checking D, we can see whether the roots are real, equal, or not real. D = b^2 - 4ac. Use the coefficients from standard form and keep the sign of b exactly as it appears in the equation. For x^2 - 4x + 1 = 0, a = 1, b = -4, c = 1. So D = (-4)^2 - 4(1)(1) = 16 - 4 = 12. Since D is positive, the equation has two real roots. Students may be asked to calculate D, interpret its value, or use D to predict the nature of roots. Writing D = 4 - 12 for 3x^2 + 2x + 7 = 0 is wrong because 4ac was not calculated correctly.
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