Multiple choice

If one root of the equation $a x^{2} +bx+c = 0$ where $a,b, c$ are integers is $ \sqrt{5} +3$, then the other root is

  1. $ \sqrt{5}  - 3$
  2. $3 -  \sqrt{5} $
  3. $ -3 - \sqrt{5} $
  4. $2 \sqrt{5} +3$
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B Correct answer
Explanation

For a quadratic equation with integer coefficients, irrational roots must occur in conjugate pairs. If one root is 3 + sqrt(5), the other must be 3 - sqrt(5).

AI explanation

Because the coefficients a, b, and c of the quadratic equation ax^2 + bx + c = 0 are rational integers, any irrational root must occur with its conjugate. Since one root is the irrational number sqrt(5) + 3, the other root must be its conjugate. The conjugate is found by negating the irrational part, giving 3 - sqrt(5).

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