Multiple choice

If one root of the equation $x ^ { 2 } + p x + q = 0$ is $2 + \sqrt { 3 },$ then the values of $p$ and $q$ are respectively.

  1. $- 4,1$
  2. $4 , - 1$
  3. $2 , \sqrt { 3 }$
  4. $- 2 - \sqrt { 3 }$
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A Correct answer
Explanation

For a quadratic equation with rational coefficients, if 2+sqrt(3) is a root, the other must be 2-sqrt(3). The sum of roots = -p = (2+sqrt(3)) + (2-sqrt(3)) = 4, so p = -4. The product of roots = q = (2+sqrt(3))(2-sqrt(3)) = 4-3 = 1.

AI explanation

Since the quadratic equation has real coefficients, the irrational conjugate 2 - sqrt(3) must be the other root. The sum of the roots equals -p, so -p = (2 + sqrt(3)) + (2 - sqrt(3)) = 4, giving p = -4. The product of the roots equals q, so q = (2 + sqrt(3))(2 - sqrt(3)) = 4 - 3 = 1. Therefore, the values of p and q are -4 and 1 respectively.

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