Solve and explain if extraneous

Question

asked Nov 25, 2024 201k views

1 Answer

Adrpino · Answer 1 · 2024-11-29T20:21:04+0000

x = 5

Yes, this solution is extraneous.

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.