Nature Of Roots

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For the quadratic equation 2x^2 - 7x + 6 = 0 with real coefficients, the discriminant analysis shows the roots have which character and form?

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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.

Nature Of Roots

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