Question

Solve and explain if extraneous

Answer 1

x = 5

Yes, this solution is extraneous.

Explanation:

Given equation:

`√(3x+1)+8=4`

Subtract 8 from both sides of the equation:

`√(3x+1)+8-8=4-8`

`√(3x+1)=-4`

Now, square both sides to eliminate the square root:

`\left(√(3x+1)\right)^2=(-4)^2`

`3x+1=16`

Subtract 1 from both sides of the equation:

`3x+1-1=16-1`

`3x=15`

Divide both sides by 3 to isolate x:

`(3x)/(3)=(15)/(3)`

`x = 5`

An extraneous solution is a solution that is produced by solving the problem, but is not a valid solution to the problem.

To check if the solution is extraneous, substitute x = 5 back into the original equation:

`√(3(5)+1)+8\overset{?}=4`

`√(16)+8\overset{?}=4`

`4+8\overset{?}=4`

`12 \\eq4`

Therefore, as the left side is not equal to the right side, the solution x = 5 is extraneous, and the equation has no solution.