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a, b, c are real numbers. The quadratic equation ax^2 - bx + c = 0 has equal roots, which is beta; then A. beta = b/c B. beta^2 = ac C. beta^3 = bc/(2a^2) D. beta^2 != b^2/(4ac)

GATE 2020 · General Aptitude · Quadratic Equations · medium

Answer: beta^3 = bc/(2a^2), so the answer is C.

  1. Apply Vieta's formulas for equal roots: Since both roots = beta: 2*beta = b/a => beta = b/(2a). Also beta^2 = c/a.
  2. Compute beta^3: beta^3 = beta * beta^2 = [b/(2a)] * [c/a] = bc/(2a^2).
  3. Verify other options are false: A: beta = b/(2a) != b/c in general. B: beta^2 = c/a != ac in general. D: beta^2 = c/a = b^2/(4a^2) by equal root condition, and b^2/(4ac) = b^2/(4a * 4a^2/b^2) is not simply b^2/(4ac) unless specific -- actually beta^2 = c/a and b^2/(4ac) = (4a^2*beta^2)/(4a*a*beta^2) = 1... verifying: beta^2 = b^2/(4a^2) from beta = b/(2a), NOT b^2/(4ac). Option D claims beta^2 != b^2/(4ac) -- since beta^2 = b^2/(4a^2), D is incorrect when c = a.