Multiple choice

The set of values of $p$ for which the roots of the equation $3x^2 + 2x + p(p - 1) = 0$ are of opposite sign is

  1. $(- \infty, 0)$
  2. $(0, 1)$
  3. $(1, \infty)$
  4. $(0, \infty)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For roots to have opposite signs, the product of roots (c/a) must be negative. Here, p(p - 1)/3 < 0. This inequality holds when p is between 0 and 1.

AI explanation

For a quadratic equation to have roots of opposite signs, the product of its roots must be strictly negative. The product of the roots for 3x squared plus 2x plus p times (p minus 1) is p times (p minus 1) divided by 3. We set the product to be negative, meaning p times (p minus 1) divided by 3 is less than 0. Multiplying by the positive 3 gives p times (p minus 1) is less than 0. This inequality holds true for values of p strictly between 0 and 1.

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