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medium importancemedium difficulty7 min

Discriminant

The discriminant of a quadratic equation ax2 + bx + c = 0 is D = b2 - 4ac.

In one line: Discriminant decides the kind of roots.

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Understand it

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.

Example

For x2 - 4x + 1 = 0, D = (-4)2 - 4(1)(1) = 12.

Formula and rule

D = b2 - 4ac.

Use the coefficients from standard form and keep the sign of b exactly as it appears in the equation.

See it worked out

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

Worked example

For x2 - 4x + 1 = 0, a = 1, b = -4, c = 1.

  1. 1So D = (-4)2 - 4(1)(1) = 16 - 4 = 12.
  2. 2Since D is positive, the equation has two real roots.

Method you can reuse

  1. 1Identify a, b, and c.
  2. 2Substitute them into D = b2 - 4ac.
  3. 3Calculate carefully with signs.
  4. 4Interpret the result if asked.

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the discriminant of 3x2 + 2x + 7 = 0?

Think first, then show answer
D = b2 - 4ac = 22 - 4(3)(7) = 4 - 84 = -80. So the discriminant is -80, and it is negative.

Common wrong answer

Writing D = 4 - 12 for 3x2 + 2x + 7 = 0.

Why it loses marks: 4ac was not calculated correctly.

Don't mix this up

Students often forget that b includes its sign and write b2 with the wrong value.

Exam tip

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

Students may be asked to calculate D, interpret its value, or use D to predict the nature of roots.
  1. 1

    Enough for 1 mark

    The discriminant of a quadratic equation ax2 + bx + c = 0 is D = b2 - 4ac.

  2. 2

    Add for 2 marks

    Example: For x2 - 4x + 1 = 0, D = (-4)2 - 4(1)(1) = 12.

  3. 3

    Add for 3 marks

    The discriminant tells us about the roots before we actually solve the equation.

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