Multiple choice

If the roots of the equation px2 + 2px + 1 = 0 are real and distinct, how many integers value(s) can p assume between 0 and 5?

  1. 0

  2. 1

  3. 3

  4. 4
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C Correct answer
Explanation

For real and distinct roots, discriminant D > 0. D = (2p)^2 - 4(p)(1) = 4p^2 - 4p > 0. p(p-1) > 0. This holds for p > 1 or p < 0. Between 0 and 5, integers are 2, 3, 4. There are 3 such integers.

AI explanation

For the quadratic equation px squared plus 2px plus 1 equals 0 to have real and distinct roots, the discriminant must be strictly greater than zero. Therefore, (2p) squared minus 4p equals 4p squared minus 4p greater than 0. This simplifies to p squared minus p greater than 0, which implies p less than 0 or p greater than 1. Since the question asks for integer values between 0 and 5, we evaluate the integers 1, 2, 3, and 4. Only the integers 2, 3, and 4 satisfy the condition of being strictly greater than 1, meaning p can assume exactly three integer values.

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