First, subtract #color(red)(3)# from each side of the equation to isolate the radical while keeping the equation balanced:
#sqrt(7r + 2) + 3 - color(red)(3) = 7 - color(red)(3)#
#sqrt(7r + 2) + 0 = 4#
#sqrt(7r + 2) = 4#
Next, square both sides of the equation to eliminate the radical while keeping the equation balanced:
#(sqrt(7r + 2))^2 = 4^2#
#7r + 2 = 16#
Then, subtract #color(red)(2)# from each side of the equation to isolate the #r# term while keeping the equation balanced:
#7r + 2 - color(red)(2) = 16 - color(red)(2)#
#7r + 0 = 14#
#7r = 14#
Now, divide each side of the equation by #color(red)(7)# to solve for #r# while keeping the equation balanced:
#(7r)/color(red)(7) = 14/color(red)(7)#
#(color(red)(cancel(color(black)(7)))r)/cancel(color(red)(7)) = 2#
#r = 2#
To validate the solution substitute #color(red)(2)# for #color(red)(r)# in the original equation and calculate the result to ensure both sides of the equation are equal (remember, the square root of a number produces a positive and negative result):
#+-sqrt(7color(red)(r) + 2) + 3 = 7# becomes:
#+-sqrt((7 * color(red)(2)) + 2) + 3 = 7#
#+-sqrt(14 + 2) + 3 = 7#
#+-sqrt(16) + 3 = 7#
#-4 + 3 = 7# and #4 + 3 = 7#
#-1 != 7# and #7 = 7#
The negative result of the radical is an extraneous solution.
The positive result shows the solution is correct.
