MHB How can you factor out x from the original term?

[tmt1]

In a text,

We have this:

$\sqrt{2x^2 + 1}$

is equal to

$x \sqrt{2 + \frac{1}{x}}$

I am confused as to how to factor out x from the original term.

 

[Euge]

The two expressions are not equal. Actually,

$$\sqrt{2x^2 + 1} = x\sqrt{2 + \frac{1}{x^2}}\quad \text{for}\quad x > 0.$$

For if $x > 0$,

$$2x^2 + 1 = 2x^2 + \frac{x^2}{x^2} = x^2\Bigl(2 + \frac{1}{x^2}\Bigr),$$

so that

$$\sqrt{2x^2 + 1} = \sqrt{x^2\Bigl(2 + \frac{1}{x^2}\Bigr)} = \sqrt{x^2} \sqrt{2 + \frac{1}{x^2}} = x\sqrt{2 + \frac{1}{x^2}}.$$

Note the use of the identity $\sqrt{x^2} = x$, $x > 0$.

More generally,

$$\sqrt{2x^2 + 1} = |x|\sqrt{2 + \frac{1}{x^2}}\quad \text{for} \quad x \neq 0$$