How to Solve Complex Quadratic Equations in Calculus?

[GreenPrint]

Homework Statement

8/25 = 1/x + ( 2 SQRT(30) )/( 13 SQRT(x^2 - 2500/169) ) [ (13 x)/50 - 50/(13 x)]

I was working on a problem from my calculus book and got this for the equation that I have to solve for x and am at a lost as to how... thanks for any help

Homework Equations

The Attempt at a Solution

 

[lanedance]

first, so we can see it a bit better
$$\frac{8}{25} <br /> = \frac{1}{x} + <br /> \frac{ 2 \sqrt{30} }{ 13 \sqrt{x^2 - \frac{2500}{169}} }<br /> ( \frac{13 x}{50} - \frac{50}{13 x })$$

 

[lanedance]

this simplification should help
$$\frac{8}{25} - \frac{1}{x} = <br /> \frac{ 2 \sqrt{30} }{ 13 \sqrt{x^2 - \frac{2500}{169}} }<br /> ( \frac{13 x}{50} - \frac{50}{13 x })$$

$$\frac{8}{25} - \frac{1}{x} = <br /> \frac{ 2 \sqrt{30} }{ 13 \sqrt{x^2 - \frac{50^2}{13^2}} }<br /> ( \frac{13 x}{50} - \frac{50}{13 x })$$

$$\frac{8}{25} - \frac{1}{x} = <br /> \frac{ 2 \sqrt{30} }{ 13 \sqrt{x^2 - \frac{50^2}{13^2}} }<br /> ( \frac{13 x}{50} - \frac{50}{13 x })(\frac{13x}{13x}\frac{50}{50})$$

$$\frac{8}{25} - \frac{1}{x} = <br /> \frac{ 2 \sqrt{30} }{\sqrt{x^2 - \frac{50^2}{13^2}} }<br /> ( x^2 - \frac{50^2}{13^2 })(\frac{1}{50x})$$

$$\frac{8}{25} - \frac{1}{x} = <br /> (\frac{ \sqrt{30}}{25x})\sqrt{x^2 - \frac{50^2}{13^2}}$$

 

[Ray Vickson]

Often in such cases we must resort to numerical methods, or approximations of some kind.

RGV

 

[lanedance]

don't think you need a numerical methods here, it should reduce to a quadratic if you continue simplifying