Multiple choice

For $a \le 0$, the roots of the equation ${x^2} - 2a\left| {x - a} \right| - 3{a^2} = 0$ is

  1. $ - a\left( {\sqrt 6 + 1} \right)$
  2. $a\left( {\sqrt 6 + 1} \right)$
  3. $a\left( {1 - \sqrt 2 } \right),a\left( { - 1 + \sqrt 6 } \right)$
  4. $a\left( {1 -\sqrt 2 } \right)$
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D Correct answer
AI explanation

For a <= 0 and testing the case where x >= a, the absolute value |x - a| becomes x - a. Substituting this into the equation yields x^2 - 2a(x - a) - 3a^2 = 0, which simplifies to x^2 - 2ax + 2a^2 - 3a^2 = 0 and factors as x^2 - 2ax - a^2 = 0. Using the quadratic formula, the roots are x = (2a +/- sqrt((-2a)^2 + 4a^2)) / 2, which gives x = a +/- a*sqrt(2). Choosing the valid root for the x >= a domain, we get a(1 - sqrt(2)) and a(1 + sqrt(2)). Since a <= 0, the root a(1 + sqrt(2)) falls outside the x >= a domain and is discarded. The result is a(1 - sqrt(2)).

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