Discriminant and Nature of Roots
For ax2 + bx + c = 0 with real coefficients, D = b2 - 4ac decides whether the roots are real and distinct, real and equal, or a conjugate pair.
What is Discriminant and Nature of Roots?
For ax2 + bx + c = 0 with real coefficients, D = b2 - 4ac decides whether the roots are real and distinct, real and equal, or a conjugate pair.
Key formula / rule: D > 0 gives two distinct real roots; D = 0 gives a repeated root; D < 0 gives conjugate complex roots.
Key points
- Compute the discriminant and classify the nature of the roots.
- Derive parameter conditions from a required nature of roots.
Common exam trap
Applying the conjugate-pair rule when the coefficients are not real.
Definitions
- Term
Discriminant and Nature of Roots
- Meaning
For ax2 + bx + c = 0 with real coefficients, D = b2 - 4ac decides whether the roots are real and distinct, real and equal, or a conjugate pair.
- Term
Discriminant and Nature of Roots — explanation
- Meaning
The discriminant is a single number that answers a qualitative question without solving the equation. In JEE it usually appears inside a parameter problem where a condition on D produces an inequality in the parameter.
Learning objectives
Compute the discriminant and classify the nature of the roots.
Derive parameter conditions from a required nature of roots.
Formulae
- Key point
D > 0 gives two distinct real roots; D = 0 gives a repeated root; D < 0 gives conjugate complex roots.
- Key point
With rational coefficients, D a perfect square means rational roots.
- Key point
Complex roots of a real quadratic always occur in conjugate pairs.
Common mistakes
Applying the conjugate-pair rule when the coefficients are not real.
Solving D ≥ 0 but forgetting the separate condition a is non-zero.
Keywords
Discriminant
Nature
Roots
Practice preview
What is the nature of the roots of the quadratic equation 2x^2 - 5x + 3 = 0?…
easy
For what value of k will the quadratic equation x^2 - 4x + k = 0 have real and equal roots?…
easy
If the discriminant of a quadratic equation is negative, what can be concluded about its roots?…
easy