We are given the equation , where and are natural numbers.
Expanding the left-hand side:
This gives us two parts: - The rational part: . - The irrational part:
Equating the rational parts and the irrational parts from both sides of the equation, we get:
, .
From the second equation, , we can solve for :
Now, substitute into the first equation:
Simplifying:
Multiply through by to clear the denominator:
Rearranging:
Let , so the equation becomes:
Solving this quadratic equation using the quadratic formula:
Thus, or . Since , we find that , so .
Now substitute into the equation :
Thus, and , so:
Therefore, the correct answer is Option (1).
If the set of all values of , for which the equation has three distinct real roots, is the interval , then is equal to
If the equation has equal roots, where and , then is equal to .