Multiple choice

If one root of the equation ax2 + bx + c = 0 is twice the other, then

  1. 2b2 = 9ac

  2. b2 = 4ac

  3. b2 = 9ac

  4. None of these
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A Correct answer
Explanation

Let the roots be alpha and 2*alpha. The sum of roots is -b/a = 3*alpha, so alpha = -b/(3a). The product of roots is c/a = 2*alpha^2 = 2*(-b/(3a))^2 = 2*b^2/(9a^2). Simplifying gives c/a = 2*b^2/(9a^2), which leads to 9ac = 2b^2.

AI explanation

Let the roots of the equation ax2 + bx + c = 0 be alpha and 2 alpha. The sum of the roots is 3 alpha = -b/a, and the product is 2 alpha2 = c/a. Squaring the sum equation gives 9 alpha2 = b2/a2, and solving the product equation for alpha2 gives alpha2 = c/2a. Substituting this back into the squared sum equation results in 9(c/2a) = b2/a2, which simplifies to 2b2 = 9ac.

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