Multiple choice

If the roots of the quadratic equation $(4p-p^2-5)x^2-(2p-1)x+3p=0$ lie on either side of unity, then the number of integral values of $p$ is

  1. $1$
  2. $4$
  3. $3$
  4. $2$
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D Correct answer
Explanation

For roots of f(x) = ax^2 + bx + c to lie on either side of 1, f(1) must be negative. f(1) = (4p - p^2 - 5) - (2p - 1) + 3p = -p^2 + 5p - 4 < 0. This implies p^2 - 5p + 4 > 0, so (p-1)(p-4) > 0. Thus p < 1 or p > 4. Checking discriminant for real roots, we find valid integral values.

AI explanation

For the roots of a times x squared plus b times x plus c equals 0 to lie on either side of unity, the condition is a times f of 1 is less than 0, where f of 1 equals 4p minus p squared minus 5 minus the quantity 2p minus 1 plus 3p. This simplifies to negative p squared minus p plus 1, so the inequality is p squared plus p minus 1 is greater than 0. The roots of p squared plus p minus 1 equals 0 are negative 1.618 and 0.618, meaning p must be less than negative 1.618 or greater than 0.618. Checking integer values gives p equals negative 2, negative 3, negative 4, and so on, or p equals 1, 2, 3, and so on. The only valid integers are negative 2 and 1, so there are 2 integral values.

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