Question

John and Tim are looking at the equation the square root of the quantity of 3 times x minus 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 equation is: \[\sqrt{3x-4} = \sqrt{x}\]

Answer 2

yes

Answer 3

So if you were to try to solve this what would your first step be?

Answer 4

I have no clue..

Answer 5

Alright, no worries. Is there anything that you don't "like" about the equation. Anything that looks "scary"?

Answer 6

all of it..

Answer 7

lol, I was afraid you would say that. :) But, don't you agree that it would be much "nicer" if it was 3x - 4 = x instead?

Answer 8

yeah, cuz then you just add four to both sides and divide by 3

Answer 9

Exactly! That would be great... except that's not the equation we have... But wait! We can grant our own wish if we just square both sides like this:
\[(\sqrt{3x-4})^2=(\sqrt{x})^2\]
\[3x - 4 = x\]

Answer 10

so then you have x=(x-4)/3

Answer 11

then what?

Answer 12

*x+4, sorry

Answer 13

Ok, so we know that x = (x + 4)/3... but that still seems really hard right? Maybe we went the wrong way... But that's okay, let's go back to the last point that was working.

Answer 14

3x - 4 = x

Answer 15

What if I instead subtracted the x from both sides?

2x - 4 = 0

and then added 4
2x = 4

Does that look better? See what you get.

Answer 16

x=2

Answer 17

so the solution is extraneous then

Answer 18

OK, you are probably right, but how do you know that it is extraneous?

Answer 19

We can always test by plugging it into the equation. If it is true it is "non-extraneous" if it is false it is "extraneous"

Answer 20

It seems to work to me: \[\sqrt{3(2)-4}=\sqrt{2}\] \[\sqrt{6-4} = \sqrt{2}\] \[\sqrt{2}=\sqrt{2}\]

Answer 21

So actually this solution looks "non-extraneous" or "true" to me.