The trick to solve these kind of problems is to take all the non-radical expressions on one side, and then square both sides to remove the radical. Let us try out our strategy for this particular problem.
sqrt( 2x + 20 ) + 2 = x
=> sqrt( 2x + 20 ) = x - 2
=> 2x + 20 = ( x - 2 )^2 (Squaring both sides)
=> 2x + 20 = x^2 - 4x + 4
=> x^2 - 6x - 16 = 0
=> x^2 - 8x + 2x - 16 = 0
=> x ( x - 8 ) +2 ( x - 8 ) = 0
=> ( x + 2 ) ( x - 8 ) = 0
=> x = -2 or x = 8
Now, we need to check the validity of our solution.
First, we check x = 8 . Putting the value in the LHS, we get 8, so it is a solution.
Next we check x = -2 . Putting the value in the LHS, we get 6. But the RHS = x = -2 . So -2 is not a valid solution of the equation. These kind of roots are called extraneous roots in literature.
