Find all x that satisfy the equation
\[ \sqrt{2x+1} + \sqrt{x-3} = 2\sqrt x.\]
+128408 First, square both sides
2x + 1 + 2sqrt [ ( 2x + 1)(x - 3) ] + x - 3 = 4x simplify
3x - 2 + 2sqrt [ ( 2x + 1)(x - 3) ] = 4x
2sqrt [ ( 2x + 1) ( x -3) ] = x + 2 square both sides again
4 ( 2x + 1) ( x - 3) = x^2 + 4x + 4
4 ( 2x^2 -5x - 3) = x^2 + 4x + 4
8x^ 2 - 20x - 12 = x^2 + 4x + 4
7x^2 - 24x - 16 = 0 factor as
(7x + 4 ) ( x -4 ) = 0
The first factor set to 0 and solved for x gives x = -4/7
This is no good because we get negatives under all the radicals with this solution
The second factor set to 0 and solved for x gives us the solution x = 4
