How Can an Expression Like ax^2 + b/x^2 + c Be a Perfect Square?

[Miike012]

how do you find the perfect square of say

ax² + b/x² + c
??

 

[Mentallic]

You can't find the perfect square of that problem precisely, unless you're satisfied with $$\left(\sqrt{ax^2+\frac{b}{x^2}+c}\right)^2$$ which I doubt since it's trivial, but take a look at the expansion of

$$\left(x+\frac{1}{x}\right)^2$$

 

[Mark44]

how do you find the perfect square of say

ax² + b/x² + c
??

You didn't by chance mean (ax² + b)/(x² + c), did you? If so, the lack of parentheses around the numerator and denominator completely confused Mentallic about what you're asking.

 

[Mentallic]

You didn't by chance mean (ax² + b)/(x² + c), did you? If so, the lack of parentheses around the numerator and denominator completely confused Mentallic about what you're asking.

That possibility completely skipped my mind

 

[Mark44]

Mentallic,
Well, I'm about as puzzled by this problem as you must be. The way I read it, the OP just wants to square the original expression, whatever it is.

 

[Miike012]

Nope that is what I ment to say.. I added an example to the paint doc and highlighted the portion in red.

It has to do with finding the surface area of a curve... and basically I was unaware the equation could be turned into a perfect square... so was wondering if there is some pattern I should look for ?

 

[Mentallic]

Mentallic,
Well, I'm about as puzzled by this problem as you must be. The way I read it, the OP just wants to square the original expression, whatever it is.

I didn't have any doubts about what the OP is trying to do, just what the expression was meant to be once you raised the point.Mike, like I was saying it doesn't work in general that ax² + b/x² + c can be turned into a perfect square, but in this case c happened to be the right number for the job.

When you get to the expression

$$\frac{25}{36}x^8+\frac{1}{2}+\frac{9}{100}x^{-8}$$

You should realize that it could be of the form $$\left(ax^4+bx^{-4}\right)^2$$ where in this case $$a=\sqrt{\frac{25}{36}}=\frac{5}{6}$$
$$b=\sqrt{\frac{9}{100}}=\frac{3}{10}$$

And all you'd need to do is check to see if $$2\cdot \frac{5}{6}\cdot \frac{3}{10} =\frac{1}{2}$$