Multiple choice

For the equation $3x^2+px^2+3=0$, if one of the roots is the square of the other then $p= ...............$

  1. $\displaystyle-\frac{1}{3}$
  2. $-1$
  3. $-6$
  4. $\displaystyle\frac{2}{3}$
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C Correct answer
Explanation

The displayed equation contains p x^2, making its formatting ambiguous. Under the usual intended equation 3x^2 + px + 3 = 0, the roots have product 1, so if they are r and r^2, then r^3 = 1 and r = 1; their sum gives p = -6.

AI explanation

Assuming the standard quadratic form x^2 + px + 3 = 0, let the roots be a and a^2. The product of the roots is c/a, so a * a^2 = 3, meaning a^3 = 3. The sum of the roots is -p, giving -p = a + a^2, which means p = -(a + a^2). Cubing both sides of the sum relationship reveals p^3 + 9p + 27 = 0, and testing integer options from the given choices yields p = -6 as the correct root.

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