First, square both sides of the equation to eliminate the square root while keeping the equation balanced:
(sqrt(6x - 20))^2 = 4^2(√6x−20)2=42
6x - 20 = 166x−20=16
Next, add color(red)(20)20 to each side of the equation to isolate the xx term while keeping the equation balanced:
6x - 20 + color(red)(20) = 16 + color(red)(20)6x−20+20=16+20
6x - 0 = 366x−0=36
6x = 366x=36
Now, divide each side of the equation by color(red)(6)6 to solve for xx while keeping the equation balanced:
(6x)/color(red)(6) = 36/color(red)(6)6x6=366
(color(red)(cancel(color(black)(6)))x)/cancel(color(red)(6)) = 6
x = 6
If we substitute color(red)(6) for color(red)(x) in the original equation and evaluate the square root we will find the extraneous solutions. Remember, the square root of a number produces a positive and negative result:
sqrt(6color(red)(x) - 20) = 4 becomes:
+-sqrt((6 * color(red)(6)) - 20) = 4
Or
-sqrt((6 * color(red)(6)) - 20) = 4 and sqrt((6 * color(red)(6)) - 20) = 4
-sqrt(36 - 20) = 4 and sqrt(36 - 20) = 4
-sqrt(16) = 4 and sqrt(16) = 4
-4 != 4 and 4 = 4
-sqrt(16) is an extraneous solution.
