Multiple choice

If one root of a quadratic equation with rational coefficient is $\begin{pmatrix}\displaystyle\frac{3\sqrt{5}}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}\end{pmatrix}$ then quadratic equation is

  1. $x^2+2x-1=0$
  2. $x^2-2x-1=0$
  3. $x^2-2\sqrt{2}+1=0$
  4. $x^2+2x-3=0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Simplify the root: 3*sqrt(5) / (sqrt(10) + 2*sqrt(5) + 2*sqrt(10) - sqrt(5) - 2*sqrt(5)*sqrt(2)) = 3*sqrt(5) / (3*sqrt(10) - sqrt(5)). Rationalizing this leads to 1 - sqrt(2). If roots are rational coefficients, the other root is 1 + sqrt(2). Sum = 2, Product = 1 - 2 = -1. Equation: x^2 - 2x - 1 = 0.

AI explanation

Simplify the denominator by recognizing that sqrt(20) = 2sqrt(5), sqrt(40) = 2sqrt(10), and sqrt(80) = 4sqrt(5). The denominator becomes 3sqrt(10) - 3sqrt(5) = 3(sqrt(10) - sqrt(5)), so the root simplifies to sqrt(5) / (sqrt(10) - sqrt(5)). Rationalizing this gives (sqrt(5)(sqrt(10) + sqrt(5))) / 5 = (sqrt(50) + 5) / 5 = (5sqrt(2) + 5) / 5 = 1 + sqrt(2). Since the coefficients are rational, the other root must be the conjugate 1 - sqrt(2), making the sum of roots 2 and the product -1, which yields the equation x^2 - 2x - 1 = 0.

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