add 8 to both sides of the equation to get sqrt(5x + 39) = 3x - 7
square both sides of the equation to get 5x + 39 = (3x - 7)^2
simplify to get 5x + 39 = 9x^2 - 42x + 49
subtract (5x + 39) from both sides of the equation to get 0 = 9x^2 - 47x + 10
factor the right side of the equation to get 0 = (x - 5) * (9x - 2)
solve for x to get x = 5 or x = 2/9
replace x in your original equation to see if these possible solutions are good.
your original equation is sqrt(5x + 39) - 8 = 3x - 15
when x = 5, this equation becomes sqrt(5*5 + 39) - 8 = 3*5 - 15.
evaluate both sides of this equation to get sqrt(64) - 8 = 15 - 15.
simplify to get 0 = 0
x = 5 is one of the solutions to your original equation.
when x = 2/9, this equation becomes sqrt(5 * 2/9 + 39) - 8 = 3 * 2/9 - 15
evaluate both sides of this equation to get sqrt(10/9 + 39) - 8 = 6/9 - 15
simplify to get 6 and 1/3 = -14 and 1/3.
this is not a true equaation, therefore x = 2/9 is not a solution to your original equation.
the solution to your original equation is x = 5.
you can see this graphically by graphing both equations.
the first equation to graph is y = 3x - 15
the second equation to graph is y = sqrt(5x+39)-8
the graph looks like this:
the intersection point of the graph of these 2 equations is only at x = 5.