For $\sqrt { 4+\sqrt { 83 } } $, the correct option is .....................
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Does not exist as quadratic surd
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Does not exist as real numbers
- $2+\sqrt { 83 } $
- $\sqrt { 83 }-2 $
A quadratic surd is of the form sqrt(a + sqrt(b)). For it to be simplified to sqrt(x) + sqrt(y), (a^2 - b) must be a perfect square. Here, 4^2 - 83 = 16 - 83 = -67, which is not a perfect square. Thus, it cannot be simplified into a simple surd form.
Assume the expression sqrt(4 + sqrt(83)) can be written as a quadratic surd of the form sqrt(a) + sqrt(b). Squaring both sides gives 4 + sqrt(83) = a + b + 2sqrt(ab). Equating the rational and irrational parts gives a + b = 4 and 2sqrt(ab) = sqrt(83). Squaring the second equation yields 4ab = 83, but substituting b = 4 - a gives 4a(4 - a) = 83, or 4a^2 - 16a + 83 = 0. The discriminant of this quadratic is (-16)^2 - 4(4)(83) = 256 - 1332, which is negative, proving the expression cannot exist as a quadratic surd.