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No Real Roots When D < 0

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    Trying to write two real numbers after D = -16 is wrong because a negative discriminant blocks real solutions.

Definition

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

Example

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

Rule to remember

When D < 0, √D is not a real number. Therefore, the quadratic equation has no real roots.

Memory hook

Negative D means no real answer.

Examples and method

Worked example

For x^2 + 2x + 5 = 0, a = 1, b = 2, c = 5. So D = 2^2 - 4(1)(5) = 4 - 20 = -16. Since D < 0, there are no real roots.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

Trying to write two real numbers after D = -16 is wrong because a negative discriminant blocks real solutions.

Exam tip

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

Quick check

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

You should conclude 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

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

2-mark answer

If the discriminant of a quadratic equation is negative, the equation has no real roots. When D < 0, √D is not a real number. Therefore, the quadratic equation has no real roots. x^2 + 2x + 5 = 0 has D = 4 - 20 = -16, so it has no real roots.

3-mark answer

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. When D < 0, √D is not a real number. Therefore, the quadratic equation has no real roots. For x^2 + 2x + 5 = 0, a = 1, b = 2, c = 5. So D = 2^2 - 4(1)(5) = 4 - 20 = -16. Since D < 0, there are no real roots. The exam may give the full equation or only the discriminant and ask whether real roots exist. Trying to write two real numbers after D = -16 is wrong because a negative discriminant blocks real solutions.
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