First, subtract color(red)(3)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)√7r+2+3−3=7−3
sqrt(7r + 2) + 0 = 4√7r+2+0=4
sqrt(7r + 2) = 4√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)2=42
7r + 2 = 167r+2=16
Then, subtract color(red)(2)2 from each side of the equation to isolate the rr term while keeping the equation balanced:
7r + 2 - color(red)(2) = 16 - color(red)(2)7r+2−2=16−2
7r + 0 = 147r+0=14
7r = 147r=14
Now, divide each side of the equation by color(red)(7)7 to solve for rr while keeping the equation balanced:
(7r)/color(red)(7) = 14/color(red)(7)7r7=147
(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.
