(x−a)(x−b) is square invariant if (x−a)(x−b) = (x−a²)(x−b²). From all square invariant quadratic polynomials chosen at random, the probability that the roots are equal is __________ (round to 1 decimal place).

Answer: 0.5Finding all square-invariant pairs (a,b): Need {a,b} = {a²,b²} as sets.Solutions:(0,0) → polynomial x² ← equal roots(0,1) → x(x−1) ← different roots(1,0) → same polynomial as (0,1)(1,1) → (x−1)² ← equal roots(ω,ω²) → (x−ω)(x−ω²) = x²+x+1 ← different roots (ω = primitive cube root of unity)(ω²,ω) → same polynomialDistinct polynomials: x², x(x−1), (x−1)², x²+x+1. Total = 4.Equal roots: x² and (x−1)² → 2 out of 4. Probability = 2/4 = 0.5.

GATE-CS is a competitive examination conducted in India. This question is from the 2025 GATE-CS Engineering Mathematics paper.

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(x−a)(x−b) is square invariant if (x−a)(x−b) = (x−a²)(x−b²). From all square invariant quadratic polynomials chosen at random, the probability that the roots are equal is __________ (round to 1 decimal place).

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