How do I find the nature of roots of a quadratic equation using the discriminant?

Question

CBSE Class 10 Quadratic Equations: questions ask to find k for which the roots are equal or real. How does the discriminant work?

GradeHunt_Team · Answer

For ax^2 + bx + c = 0 (a not equal to 0), the discriminant is D = b^2 - 4ac. Its sign tells you the nature of the roots: - D > 0: two distinct real roots - D = 0: two equal real roots, each equal to -b/2a - D < 0: no real roots Type 1: find the nature of roots. 2x^2 - 4x + 3 = 0: D = 16 - 24 = -8 < 0, so there are no real roots. Type 2: find k so that the roots are equal. kx^2 + 6x + 1 = 0 Equal roots means D = 0: 36 - 4k = 0, so k = 9. Type 3: find k so that the roots are real. x^2 + kx + 4 = 0 Real roots means D >= 0: k^2 - 16 >= 0, so k <= -4 or k >= 4. Common mistakes: 1. Using D > 0 when the question says "real roots". Real roots include equal roots, so the condition is D >= 0. 2. Forgetting that a cannot be 0. If the coefficient of x^2 contains k, reject any value of k that makes it zero. 3. Sign errors when b is negative: (-4)^2 is 16, not -16. Exam tip: always write "For equal roots, D = 0" (or the correct condition) as the first line. It shows your method clearly, and CBSE marking schemes split the marks across the steps of a solution.

How do I find the nature of roots of a quadratic equation using the discriminant?

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