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medium importancemedium difficulty7 min

No Real Roots When D < 0

If the discriminant of a quadratic equation is negative, the equation has no real roots.

In one line: Negative D means no real answer.

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Understand it

A negative discriminant means the square-root part in the quadratic formula cannot produce a real number. So the equation does not have real solutions.

Example

x2 + 2x + 5 = 0 has D = 4 - 20 = -16, so it has no real roots.

Formula and rule

When D < 0, √D is not a real number.

Therefore, the quadratic equation has no real roots.

See it worked out

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

Worked example

For x2 + 2x + 5 = 0, a = 1, b = 2, c = 5.

  1. 1So D = 22 - 4(1)(5) = 4 - 20 = -16.
  2. 2Since D < 0, there are no real roots.

Method you can reuse

  1. 1Calculate D.
  2. 2Check whether D is less than zero.
  3. 3If yes, state that there are no real roots.
  4. 4Do not continue with real-number solving.

Visual tip

A graph sketch that does not cut the x-axis can be used to show the no-real-root case.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What should you conclude when D = -16 for a quadratic equation?

Think first, then show answer
You should conclude that the equation has no real roots. A negative discriminant means the roots are not real numbers.

Common wrong answer

Trying to write two real numbers after D = -16.

Why it loses marks: A negative discriminant blocks real solutions.

Don't mix this up

Students sometimes try to find real answers even after getting a negative discriminant.

Exam tip

If D is negative, state clearly that the equation has no real roots and stop there unless complex roots are asked.

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

The exam may give the full equation or only the discriminant and ask whether real roots exist.
  1. 1

    Enough for 1 mark

    If the discriminant of a quadratic equation is negative, the equation has no real roots.

  2. 2

    Add for 2 marks

    Example: x2 + 2x + 5 = 0 has D = 4 - 20 = -16, so it has no real roots.

  3. 3

    Add for 3 marks

    A negative discriminant means the square-root part in the quadratic formula cannot produce a real number.

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