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Is It Rational? (Posted on 2023-09-09) Difficulty: 3 of 5
Solve the equation

x2(7 - √(7x2 - 3) - √(x - 1)) = x + 2

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts Solution Comment 1 of 1
Using essentially a guess-and-check approach ...
There is a chance if both (x - 1) and (7x^2 - 3) are perfect squares.
squares:  {0, 1, 4, 9, 16, ...}
from x-1, try x values:  {1, 2, 5, 10, 17, ...}
7x^2 - 3 values: {4, 25, 172, 697, ...}
x=1 works to make LHS rational, but it does not solve the equation:
    LHS=5 but RHS=3
x=2 works; and it does solve the equation:
    LHS=RHS=4

x=2 is a solution, though perhaps not the only solution.

If the two radical terms had opposite signs, there might be an x value that would allow them to cancel each other out, but since there is a negative sign in front of each √, this cannot happen without complex numbers.

Running the program below shows no other x values where the two radicals are integers

Checking Wolfram Alpha shows that x=2 is the only real solution; and there are 4 complex solutions which I did not attempt to solve for.

-----------------------
def issquare(n):
    """ Input an integer, Returns True iff it is a perfect square. """
    if round(n**0.5)**2 == n:
        return True
    else:
        return False
    
xvalues = [i**2 + 1 for i in range(-100000,100000)]
xvalues = sorted(list(set(xvalues)))

for x in xvalues:
    if issquare(7*x**2 - 3):
        print(x, (x-1)**.5, (7*x**2 - 3)**.5)
        print((x**2)*(7 -(7*x**2 - 3)**.5 - (x-1)**.5), x+2, '\n')

Output:
1 0.0 2.0
5.0 3 

2 1.0 5.0
4.0 4 

  Posted by Larry on 2023-09-09 13:44:37
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