The quadratic equation whose root is $\sqrt{2}+3$
- $x^{2}-6x+7=0$
- $x^{2}-9=0$
- $x^{2}-3x-\sqrt{2}=0$
- $x^{2}-5x+3=0$
Reveal answer
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A
Correct answer
Explanation
If a root is 3 + sqrt(2), the other root must be 3 - sqrt(2). The sum of roots is 6 and the product is (3+sqrt(2))(3-sqrt(2)) = 9 - 2 = 7. The quadratic equation is x^2 - (sum)x + (product) = 0, which is x^2 - 6x + 7 = 0.
AI explanation
Assuming rational coefficients, if one root is 3 plus the square root of 2, the other must be its conjugate, 3 minus the square root of 2. The sum of the roots is 6, and the product is 3 squared minus the square root of 2 squared, which equals 9 minus 2, or 7. Substituting these into the standard quadratic form gives the equation x squared minus 6x plus 7 equals 0.