Taking negative/positive square roots

[myink]

Let's say that the variable 'x' is definitely some negative number.

So if I wanted to solve:

$$x^2 = 4$$

I get:

$$\pm \sqrt{x^2} = \pm \sqrt{4}$$
$$\pm x = \pm 2$$

I would have to take the positive value of 'x' and the negative value of '2' to make this true...is it okay to only take a positive square root of one side of the equation and the negative square root of the other?

 

[Muphrid]

Yeah, you only put the plus or minus on one side or the other. It doesn't matter which side because either way gives you the same result. If you put it on both sides, the usual reading of the plus or minus implies plus corresponds with plus, minus with minus, and that's not what you need to happen. It's more like you have four cases:

$$+x = +2 \quad \text{ or } +x = -2 \quad \text{ or } -x = +2 \quad \text{ or } -x = -2$$

Two of those are redundant, and you can get enough information out of the equation with one plus or minus sign to do the job.

 

[myink]

Oh, thank you. I was just accustomed to learning that if you perform one operation to one side of the equation, you have to do the equivalent on the other so I thought plus/minus square root on one side would mean I have to do exactly plus/minus square root to the other. I guess this is some sort of exception, as you explained it?

 

[tiny-tim]

hi myink!

I was just accustomed to learning that if you perform one operation to one side of the equation, you have to do the equivalent on the other …

but ± isn't an operation, it's two operations …

an equation with ± in it is really two different equations, written as one to save space!

btw, Muphrid is completely correct …

If you put it on both sides, the usual reading of the plus or minus implies plus corresponds with plus, minus with minus, and that's not what you need to happen.

… "± x = ±2" means "x = 2 or -x = -2", it doesn't allow for x = -2

 

[Feodalherren]

nvm I was beaten to it :)