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 product of roots is

  1. $2$
  2. $1$
  3. $-3$
  4. $3$
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A Correct answer
Explanation

For x^2 + 3ax + 2a^2 = 0, the sum of roots alpha + beta = -3a and the product alpha*beta = 2a^2. Given alpha^2 + beta^2 = 5, we use (alpha + beta)^2 - 2(alpha*beta) = 5, which is (-3a)^2 - 2(2a^2) = 5, leading to 9a^2 - 4a^2 = 5, so 5a^2 = 5, a^2 = 1. The product of roots is 2a^2 = 2(1) = 2.

AI explanation

Using the relation between roots and coefficients, the sum of the roots is a + b = -3a and the product of the roots is ab = 2a^2. We square the sum to get (a + b)^2 = a^2 + b^2 + 2ab, which means (-3a)^2 = 5 + 2(2a^2). This simplifies to 9a^2 = 5 + 4a^2, yielding 5a^2 = 5 and thus a^2 = 1. The required product of the roots is ab = 2a^2 = 2(1) = 2.

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