If $x=3-2\sqrt{2}$, find $\sqrt{x}-\frac{1}{\sqrt{x}}$. Two methods give two answers ($-2$ vs $\pm2$).

Question

Given $x=3-2\sqrt{2}$ find value of $\sqrt{x}-\frac{1}{\sqrt{x}}$

Method 1.

Let $$\sqrt{x}-\frac{1}{\sqrt{x}}=t \tag{1}$$

Squaring both sides we get $$t^2=x+\frac{1}{x}-2=(3+2\sqrt{2})+(3-2\sqrt{2})-2=4 \tag{2}$$ So $$t=\pm 2 \tag{3}$$

Method 2.

$$\sqrt{x}=\sqrt{3-2\sqrt{2}}=\sqrt{2}-1 \tag{4}$$

So $$\sqrt{x}-\frac{1}{\sqrt{x}}=\sqrt{2}-1-(\sqrt{2}+1)=-2 \tag{5}$$

What is correct?

What value has $x = 1+1$? Squaring gives $x^2 = (1+1)^2 = 1 + 2 + 1 = 4$. So $x = \pm 2$? — Martin R, May 14 '20 at 09:27
Both are correct. Method $1$ just gives less, i.e., only $t^2=4$, which is correct. — Dietrich Burde, May 14 '20 at 09:27
Aside from what everyone else has mentioned, it is also worth noting that you have an expression entirely in terms of a constant $x$. That expression could only have one value. Imagine if $\sqrt x - 1/\sqrt x$ had two values instead. Then
$$-2 = \sqrt x - \frac 1 {\sqrt x} = 2$$

but $2 \ne -2$ obviously. — PrincessEev, May 14 '20 at 09:32
$t^2 = 4 \implies t = \pm 2$ means $t$ can be one of $+2$ or $-2$. It doesn't force $t$ to $+2$. — achille hui, May 14 '20 at 09:32
Does this answer your question? Why can't you square both sides of an equation? — Martin R, May 14 '20 at 09:49

score 4 · Accepted Answer · answered May 14 '20 at 09:30

4

Since $x=3-2\sqrt 2 \lt 1, \sqrt x \lt 1$. (the square root function is increasing) This means $\frac{1}{\sqrt x} \gt 1 \gt \sqrt x$ and so $$\sqrt x - \frac{1}{\sqrt x} \lt 0$$

answered May 14 '20 at 09:30

Vishu

14,469

score 3 · Answer 2 · answered May 14 '20 at 09:26

3

Method $2$ is correct. In method $1$, you introduced an extraneous solution when you squared both sides.

For example, $x=1\implies x^2=1\implies x=\pm1$, but the implication doesn’t work in reverse.

answered May 14 '20 at 09:26

J. W. Tanner

60,406

score 2 · Answer 3 · answered May 14 '20 at 09:27

2

Squaring will lose the information about the sign, see Why can't you square both sides of an equation?.

(P.S. You can check your results in wolfram alpha.)

answered May 14 '20 at 09:27

Vepir

12,516

If $x=3-2\sqrt{2}$, find $\sqrt{x}-\frac{1}{\sqrt{x}}$. Two methods give two answers ($-2$ vs $\pm2$).

3 Answers3