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Find all possible integer solutions for the root equation. Sqrt x + Sqrt y = Sqrt 180. Find how many pairs for (x, y) are possible.
Solution:
Given equation:
\[\sqrt{x}+\sqrt{y} = \sqrt{180}\]
\[\Rightarrow \sqrt{x}+\sqrt{y} = 6\sqrt{5}\]
5 is a prime number so x and y must contain 5:
Let a, b are coefficients as given below:
\[\Rightarrow a\sqrt{5}+b\sqrt{5} = 6\sqrt{5}\]
Now,
\[a+b=6\]
For integer solutions,
(a, b) = (0, 6), (1, 5),....., (5, 1), (6, 0).
There are infinitely many pairs possible, but here we will focus for positive solutions.
Put these values under the root with 5 to get x, y: (x, y) = (5a², 5b²):
(x, y) = (0, 180), (5, 125), (20, 80), (45, 45), (80, 20), (125, 5), (180, 0)
Total 7 Solution are possible for this equation.
(Note: √x can be -|√x| so infinitely many solutions are possible)
Question related to Algebra, Equation.
