Multiple choice

Quadratic equation whose one of the roots is $4+\sqrt{5}$ is:

  1. $x^{2}+8x-1=0$
  2. $x^{2}+8x+1=0$
  3. $x^{2}-8x+1=0$
  4. $x^{2}-8x+11=0$
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D Correct answer
Explanation

If a quadratic equation with rational coefficients has a root 4 + sqrt(5), its conjugate 4 - sqrt(5) must also be a root. The sum of the roots is (4 + sqrt(5)) + (4 - sqrt(5)) = 8, and the product is (4 + sqrt(5))(4 - sqrt(5)) = 16 - 5 = 11. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 8x + 11 = 0.

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