Paul J Nahin. Story of minus one, I'm stumped on page 5.

Question

every time I try to read this book and follow the logic I fail on just page 5 where it states near the bottom

$${1 \over x } + 14 x + \sqrt{{1\over x^2} + 196 x^2 } = 12,$$ which is easily put into the form given above, $$ 172 x = 336 x^2 +24.$$

(From An Imaginary Tale, The Story of $\sqrt{-1}$ )

But try as I may I cannot manipulate the square root equation into the quadratic stated, easy indeed! A lifetime working an an electrical engineer using j notation has been of no help to me. Please put me right if you are able. Thank you.

Answer 1

Let $\frac{1}{x}+14x=y$ then we can rewrite the equation as $y+\sqrt{y^2-28}=12 \rightarrow \sqrt{y^2-28}=12-y$

Squaring both parts we get: $y^2-28=144-24y+y^2$ or $24y=172$ or $$24(\frac{1}{x}+14x)=172$$ Can you finish?

Answer 2

The trick here is to isolate the square root on one side and then squaring both sides to get rid of it, yielding:

$$\frac{1}{x^2}+196x^2=\left(12-14x-\frac{1}{x}\right)^2$$

You can then expand the RHS and simplify to get the required form.