Multiple choice

If α and β are the roots of the equation | x |2 + | x | - 6 = 0, which of the given options is correct?

  1. α + β = 0

  2. α = β

  3. α + β = 1

  4. α2 = β
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A Correct answer
Explanation

Let y = |x|. y^2 + y - 6 = 0 => (y+3)(y-2) = 0. Since y = |x| >= 0, y = 2. So |x| = 2, meaning x = 2 or x = -2. The roots are 2 and -2. Thus, alpha + beta = 2 + (-2) = 0.

AI explanation

Let y equal the absolute value of x, which transforms the equation into y squared plus y minus six equals zero. Factoring this quadratic gives the quantity y plus three times the quantity y minus two equals zero, meaning y equals two or y equals negative three. Since y represents an absolute value, y cannot be negative, so y must equal two. The roots for x are therefore positive two and negative two. The sum of these roots, alpha plus beta, is two plus negative two, which equals zero.

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