Question

A quadratic polynomial \( (x - \alpha)(x - \beta) \) over complex numbers is said to be square invariant if \[ (x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2). \] Suppose from the set of all square invariant quadratic polynomials we choose one at random. The probability that the roots of the chosen polynomial are equal is ___________. (rounded off to one decimal place) 

Hint:

For probability in algebraic structures, equate coefficients carefully and analyze valid root conditions systematically.

Answer 1

Given that the quadratic polynomial satisfies the square invariance property: \[ (x - \alpha)(x - \beta) = (x - \alpha^2)(x - \beta^2). \] Expanding both sides, we get: \[ x^2 - (\alpha + \beta)x + \alpha\beta = x^2 - (\alpha^2 + \beta^2)x + \alpha^2\beta^2. \] By equating coefficients, we obtain the equations: \[ \alpha + \beta = \alpha^2 + \beta^2, \] \[ \alpha\beta = \alpha^2\beta^2. \] Step 1: Solving for equal roots For equal roots, we assume \( \alpha = \beta \). Substituting in the first equation: \[ 2\alpha = 2\alpha^2 \Rightarrow \alpha (1 - \alpha) = 0. \] Thus, \( \alpha = 0 \) or \( \alpha = 1 \). 
Step 2: Probability Calculation Among all possible values of \( \alpha, \beta \) satisfying the quadratic constraints, half of them lead to equal roots. Therefore, the required probability is: \[ \frac{1}{2} = 0.5. \]