Multiple choice

if the roorts of the equation $ax^{2}-4x+a^{2}=0$ are imaginary and the sum of the roots is equal to their product hen $a$ is:

  1. $-2$
  2. $4$
  3. $2$
  4. $None\ of\ these$
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C Correct answer
Explanation

The sum of the roots is 4/a and their product is a. Equating them gives a^2 = 4, so a = 2 or -2. The roots are imaginary only when the discriminant 16 - 4a^3 is negative, which holds for a = 2, not a = -2.

AI explanation

For the quadratic equation ax^2 - 4x + a^2 = 0, the sum of the roots is 4/a and the product of the roots is a^2/a = a. Because the sum equals the product, we set 4/a = a, which gives a^2 = 4, yielding possible values of a = 2 or a = -2. The condition that the roots are imaginary requires the discriminant to be less than zero, so evaluating (-4)^2 - 4(a)(a^2) < 0 gives 16 - 4a^3 < 0. Testing a = 2 makes the discriminant 16 - 32 = -16 (which is imaginary), while a = -2 makes it 48 (which is real), so a must be 2.

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