sqrt(x-8)+sqrt(x+3) = 1
=> (sqrt(x-8)+sqrt(x+3))^2 = 1^2
=> (x-8) + 2sqrt((x-8)(x+3)) + (x+3) = 1
=> 2sqrt(x^2-5x-24) = -2x+6
=> sqrt(x^2-5x-24) = -x+3
=>(sqrt(x^2-5x-24))^2 = (-x+3)^2
=>x^2-5x-24 = x^2-6x+9
=> x = 33
In the process of squaring, we may have generated extraneous solutions, and so we must check to see if our possible solution is valid.
sqrt(33-8)+sqrt(33+3) = sqrt(25)+sqrt(36) = 5+6 = 11
As the only possible value for x we found is not a solution, there is no solution. As another way of seeing this, note that the graph of
sqrt(x-8)+sqrt(x+3)-1
never intersects the x axis, and thus the equivalent equation
sqrt(x-8)+sqrt(x+3)-1 = 0
has no solutions.
graph{sqrt(x-8)+sqrt(x+3)-1 [-3.92, 76.08, -4, 36]}
