Multiple choice

If one root of quadratic equation $2{ x }^{ 2 }-px+q=0$ is $2+\sqrt { 3}\ (p,q)\in$ rational numbers , then $\dfrac { p }{ q }$ is equal to

  1. 2

  2. 3

  3. 4

  4. 8
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C Correct answer
Explanation

If coefficients are rational, irrational roots occur in conjugate pairs. Roots are 2+sqrt(3) and 2-sqrt(3). Sum of roots = 4 = p/2, so p=8. Product of roots = (2+sqrt(3))(2-sqrt(3)) = 4-3 = 1 = q/2, so q=2. p/q = 8/2 = 4.

AI explanation

Because the coefficients are rational and one root is 2 + sqrt(3), the other root must be its conjugate, 2 - sqrt(3). Using the sum of roots formula, p = (2 + sqrt(3)) + (2 - sqrt(3)) = 4, and using the product of roots formula, q = (2 + sqrt(3))(2 - sqrt(3)) = 4 - 3 = 1. Therefore, p/q = 4/1 = 4.

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