Multiple choice

$x^2+3ax+2a^2=0$ If the above equation has roots $\alpha ,\beta $ and it is given that $\alpha^2 +\beta ^2=5$, then the value of $a$ is

  1. $1$
  2. $-1$
  3. $\pm 1$

  4. None of these
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C Correct answer
Explanation

Roots alpha, beta of x^2 + 3ax + 2a^2 = 0. Sum = -3a, Product = 2a^2. alpha^2 + beta^2 = (alpha+beta)^2 - 2(alpha*beta) = (-3a)^2 - 2(2a^2) = 9a^2 - 4a^2 = 5a^2. Given 5a^2 = 5, so a^2 = 1, a = +/- 1.

AI explanation

Using the identity for the sum of squares of the roots, alpha^2 + beta^2 = (alpha + beta)^2 - 2*alpha*beta. From the equation x^2 + 3ax + 2a^2 = 0, the sum of the roots is -3a and the product is 2a^2. Substituting these values gives (-3a)^2 - 2(2a^2) = 5, which simplifies to 9a^2 - 4a^2 = 5, or 5a^2 = 5. Therefore, a^2 = 1, which means the value of a is +1 or -1.

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