Equal Roots Condition

Equal roots of a quadratic require the discriminant to vanish, the defining JEE Advanced condition for a repeated root. For (m + 1)x^2 + 2(m + 3)x + (m + 8) = 0, the discriminant is [2(m + 3)]^2 - 4(m + 1)(m + 8) = 0. Expanding, 4(m + 3)^2 - 4(m + 1)(m + 8) = 0, divide by 4: (m^2 + 6m + 9) - (m^2 + 9m + 8) = 0, giving 6m + 9 - 9m - 8 = -3m + 1 = 0, so m = 1/3. We require m + 1 ≠ 0 for a genuine quadratic, and m = 1/3 satisfies that. Option m = -1 makes the equation linear, not quadratic. Option m = 0 from sum of roots is irrelevant to equal roots. Option no real m exists contradicts the linear discriminant equation. Hence m = 1/3. Plausibility check: substituting m = 1/3 makes the discriminant exactly zero by construction, guaranteeing a single repeated real root, consistent with the equal-roots requirement. Equating the discriminant to zero is the precise algebraic statement that a parabola touches the x-axis at a single repeated root, geometrically a tangency condition. The parametric setup must also exclude any value that would destroy the leading coefficient and collapse the equation to a linear one, a subtlety that examiners deliberately embed to test whether the quadratic structure has been preserved.