Multiple choice

Find the value of '$p$' so that equation $\displaystyle { 4x }^{ 2 }-8px+9=0$ has roots whose difference is $4$.

  1. $\displaystyle \pm 3$
  2. $\displaystyle \pm \dfrac25$
  3. $\displaystyle \pm \dfrac52$

  4. None of these
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C Correct answer
Explanation

For 4x^2 - 8px + 9 = 0, roots alpha and beta satisfy: alpha + beta = 8p/4 = 2p, alpha * beta = 9/4. Given |alpha - beta| = 4. (alpha - beta)^2 = (alpha + beta)^2 - 4 * alpha * beta. 4^2 = (2p)^2 - 4 * (9/4). 16 = 4p^2 - 9. 25 = 4p^2. p^2 = 25/4. p = +/- 5/2.

AI explanation

Let the roots be a and b, so their difference a - b = 4, which means (a - b)^2 = 16. Expanding this gives a^2 - 2ab + b^2 = 16, and adding 4ab to both sides yields (a + b)^2 = 16 + 4ab. For the equation 4x^2 - 8px + 9 = 0, Vieta's formulas give the sum of roots a + b = 8p / 4 = 2p and the product ab = 9 / 4. Substituting these into the previous relation gives (2p)^2 = 16 + 4 * (9/4), so 4p^2 = 25 and p = 5/2 or p = -5/2.

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