Simplifying Rationals with Radicals

[jbolt]

First post so bare with me a moment...

I have a problem with solving a simple equation...

x/ sqrt(x^2-9) when x= -3

Obvious if you use direct substitution this would work out to -3/0 which is undefined...

If you multiply the denominator by the same sqrt(x^2-9) to remove the radical you still end up with the same undefined problem... what would be the next step into simplifying this expression? Would you then multiply by the congruate x^2 + 9?

Any help would be greatly appreciated!

 

[galactus]

This is not an equation, it is an expression. Equations have equals signs and you solve for a variable.

In this case, it is an expression because we just make a substitution into x.

As for simplifying any further, not much to do.

If we multiply top and bottom by \(\displaystyle \sqrt{x^{2}-9}\).

\(\displaystyle \frac{x}{\sqrt{x^{2}-9}}\cdot\frac{\sqrt{x^{2}-9}}{\sqrt{x^{2}-9}}\)

\(\displaystyle \frac{x\sqrt{x^{2}-9}}{x^{2}-9}\)

The denominator is rationalized. Is that what you mean?. Not much more to do.

You could factor if you like:

\(\displaystyle \frac{x}{\sqrt{(x+3)(x-3)}}\)

Multiplying top and bottom by \(\displaystyle x^{2}+9\) will not do much to help.

 

[jbolt]

That is the conclusion that I came to also. But can this be changed to represent an expression that is not undefined?

 

[BigGlenntheHeavy]

\(\displaystyle Let \ f(x) \ = \ \frac{x}{\sqrt{x^2-9}}, \ then \ f(x) \ is \ undefined \ when \ x \ = \ \pm \ 3 \ whether\)

\(\displaystyle we \ rationalized \ the \ denominator \ or \ not. \ Also \ x^2-9 \ \ge \ 0 \ \implies \ x^2 \ \ge \ 9,\)

\(\displaystyle \sqrt{x^2} \ \ge \ \sqrt 9 \ \implies \ |x| \ \ge \ 3 \ implies \ |x| \ > \ 3, \ hence \ x \ >3 \ or \ x \ < \ -3\)

\(\displaystyle (x \ is \ undefined \ when \ x \ = \ \pm \ 3) \ hence \ domain \ of \ f(x) \ is \ (-\infty,-3)U(3,\infty), \ and \ range\)

\(\displaystyle is \ (-\infty,-1)U(1,\infty).\)

\(\displaystyle x \ = \ \pm \ 3 \ are \ vertical \ asymptotes, \ y \ = \ \pm1 \ are \ horizontal \ asymptotes \ (not \ shown).\)

\(\displaystyle Note: \ That's \ all \ she \ wrote, \ the \ pencil \ broke.\)

[attachment=0:2aesm43t]aaa.jpg[/attachment:2aesm43t]

 

[Denis]

...so bare with me a moment...

Too cold for that :roll: