Finding roots of equation before integrating: x^2 + 4/(x^2) = 5

[jonnburton]

I wondered if anyone could point me in the right direction with this question:

" a) Find the positive roots of the equation [math]x^2 +\frac {4}{x^2}=5[/math]"
"b) Find the area of the region in the first quadrant, bounded by the curve [math]y+x^2 + \frac{4}{x^2}[/math] and the line y = 5"

I am stuck at part a): I can see this expression is equivalent to [math]x^2 +4x^{-\frac{1}{2}}-5 = 0[/math] but cannot think how to factorise this and obtain the roots. Any pointers would be gratefully received.

 

[fresh_42]

I wondered if anyone could point me in the right direction with this question:

" a) Find the positive roots of the equation [math]x^2 +\frac {4}{x^2}=5[/math]"
"b) Find the area of the region in the first quadrant, bounded by the curve [math]y+x^2 + \frac{4}{x^2}[/math] and the line y = 5"

I am stuck at part a): I can see this expression is equivalent to [math]x^2 +4x^{-\frac{1}{2}}-5 = 0[/math] but cannot think how to factorise this and obtain the roots. Any pointers would be gratefully received.

Multiply the equation in a) by [imath] x^2, [/imath] then substitute [imath] u=x^2 [/imath] and solve for [imath] u. [/imath]

 

[khansaheb]

I am stuck at part a): I can see this expression is equivalent to x2+4x−12−5=0x^2 +4x^{-\frac{1}{2}}-5 = 0x2+4x−21−5=0 but cannot think

That is incorrect.
Follow the hint suggested in response #2

 

[Dr.Peterson]

a) Find the positive roots of the equation [math]x^2 +\frac {4}{x^2}=5[/math]b) Find the area of the region in the first quadrant, bounded by the curve [math]y+x^2 + \frac{4}{x^2}[/math] and the line y = 5

I assume the equation in (b) is really [imath]y{\color{Red} =}x^2 + \frac{4}{x^2}[/imath]

 

[Steven G]

I wondered if anyone could point me in the right direction with this question:

" a) Find the positive roots of the equation [math]x^2 +\frac {4}{x^2}=5[/math]"
"b) Find the area of the region in the first quadrant, bounded by the curve [math]y+x^2 + \frac{4}{x^2}[/math] and the line y = 5"

I am stuck at part a): I can see this expression is equivalent to [math]x^2 +4x^{-\frac{1}{2}}-5 = 0[/math] but cannot think how to factorise this and obtain the roots. Any pointers would be gratefully received.

Don't you just see that if x^2 = 1 or 4 then 1 + 4 = 5. So x^2 = 1, then x = +/- 1,2. Since they want only the positive solution, x=1 or x=2.

Having said this, you should be able to solve this without 'seeing' the answer (although, you should see the answer!)

 

[jonnburton]

Thanks for the replies; yes, perhaps I should have seen what Steven G pointed out but I did it by substituting u=x^2 as per Fresh_42's advice.

(and apologies for the typing errors in the equations)