Question

John and Tim are looking at the equation the square root of the quantity of 3 times x plus 4 equals square root of x . John says that the solution is extraneous. Tim says the solution is non-extraneous. Is John correct? Is Tim correct? Are they both correct? Justify your response by solving this equation, explaining each step with complete sentences.

Answer 1

The only way to know is to check it
$$
\large{
\sqrt{3x+4}=x\\
3x+4=x^2\\
x^2-3x-4=0\\
(x-4)(x+1)=0
}
$$
Answer is x=4 and x=-1.

But how can the square root of something (1st equation) equal to -1 (assuming we are dealing with real numbers, not complex). Plugging in -1 into the original equation also does not work. So the only solution is x=4, x=-1 is extraneous.

The problem is a dilemma because in the second step we squared both sides, introducing -1 as a possible solution, where initially it was not.

Answer 2

So in a sense they are both correct.