The notation "sqrt" is shorthand for "square root"
sqrt(-x+10)-x = 2 sqrt(-x+10) = x+2 ... add x to both sides -x+10 = (x+2)^2 ... square both sides -x+10 = x^2+4x+4 ... use FOIL x^2+5x-6 = 0 ... get everything to one side; combine like terms (x-1)(x+6) = 0 ... factor x-1=0 or x+6 = 0 ... zero product property x = 1 or x = -6 .... solve each sub-equation for x
The possible solutions are x = 1 or x = -6. We need to check each of those solutions
Let's check x = 1 sqrt(-x+10)-x = 2 sqrt(-1+10)-1 = 2 sqrt(9)-1 = 2 3-1 = 2 2 = 2 So x = 1 has been confirmed
Let's check x = -6 sqrt(-x+10)-x = 2 sqrt(-(-6)+10)-(-6) = 2 sqrt(6+10)+6 = 2 sqrt(16)+6 = 2 4+6 = 2 10 = 2 The last equation is false, so x = -6 is extraneous. It's not a true solution
