Squaring both sides of an equation: To include or not to include the minus sign?

[phospho]

say the i had the equation x -3√x - 16 = 0, if I was to square every term to get rid of the root would it be:

1) x + 9x + 16 = 0
2) x - 9x - 16 = 0

Just confused if I should include the minus sign

 

[Simon Bridge]

You have to square the whole side - so you should have an x² in there too.
You don't get to square each term by themselves - you have to square the whole thing.

note: 5-(3+2)=0

square each term

25 - (9+4) = 0 nope
25 + (9+4) = 0 nope

see? both are wrong.

in general, just because a+b=c it does not follow that a²+b²=c²

To get rid of the root, isolate it on the LHS before squaring both sides.

 

[Feodalherren]

You should isolate your root term so that you get

3sqrt x = x-16

So when you square both sides you get

(3sqrt x)^2 = (x-16)(x-16)

You could square the whole side too as Simon pointed out but it's more laborious.

 

[micromass]

Of course, another option is to do the substitution y=\sqrt{x} and then solve for y.

 

[Mark44]

say the i had the equation x -3√x - 16 = 0, if I was to square every term to get rid of the root would it be:

1) x + 9x + 16 = 0
2) x - 9x - 16 = 0

Just confused if I should include the minus sign

If you're going to square the whole side, you get (x -3√x - 16)(x -3√x - 16). You can't just look at one term and square it, and ignor the other terms on that side. In this case, there are six multiplications that have to be done if you take this approach. You did one.

 

[Simon Bridge]

Y
(3sqrt x)^2 = (x-16)(x-16)

You could square the whole side too as Simon pointed out but it's more laborious.

... and in your example, you have squared the whole side (for both sides). That's what I meant: see last line post #2 :)

I had hoped to lead phospho to do the actual manipulation... oh well.