Nature Of Roots
For the quadratic equation 2x^2 - 7x + 6 = 0 with real coefficients, the discriminant analysis shows the roots have which character and form?
Select the correct option:
Solution
Real, distinct and rational
The discriminant D = b^2 - 4ac determines the nature of the roots of a quadratic, the first tool applied in JEE Advanced root analysis. For 2x^2 - 7x + 6 = 0, we have a = 2, b = -7, c = 6, so D = 49 - 4·2·6 = 49 - 48 = 1. Since D > 0 the roots are real and distinct, and because D = 1 is a perfect square with rational coefficients, the roots are rational. Solving, x = (7 ± 1)/4 gives x = 2 and x = 3/2, both rational and unequal. Option real equal would require D = 0. Option complex conjugates would require D < 0. Option real distinct irrational would require D positive but not a perfect square. Hence the roots are real, distinct, and rational. Plausibility check: substituting x = 2 gives 8 - 14 + 6 = 0 and x = 3/2 gives 4.5 - 10.5 + 6 = 0, both satisfying the equation, confirming the rational distinct roots.
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About This Question
- Subject
- mathematics
- Chapter
- complex numbers and quadratic equations
- Topic
- nature of roots
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
Real, distinct and rational
The discriminant D = b^2 - 4ac determines the nature of the roots of a quadratic, the first tool applied in JEE Advanced root analysis. For 2x^2 - 7x + 6 = 0, we have a = 2, b = -7, c = 6, so D = 49 - 4·2·6 = 49 - 48 = 1. Since D > 0 the roots are real and distinct, and because D = 1 is a perfect square with rational coefficients, the roots are rational. Solving, x = (7 ± 1)/4 gives x = 2 and x = 3/2, both rational and unequal. Option real equal would require D = 0. Option complex conjugates would require D < 0. Option real distinct irrational would require D positive but not a perfect square. Hence the roots are real, distinct, and rational. Plausibility check: substituting x = 2 gives 8 - 14 + 6 = 0 and x = 3/2 gives 4.5 - 10.5 + 6 = 0, both satisfying the equation, confirming the rational distinct roots.
This easy difficulty mathematics question is from the chapter complex numbers and quadratic equations, covering the topic of nature of roots. It appeared in the 2025 exam.
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