Question

if x=1/(x-5), find x^2-1/x^2

Answer 1

Hey friend, Harish here.

Here is your answer:

$x = \frac{1}{x-5}$

⇒ $(x)(x-5) = 1$

⇒ $x^{2} - 5x = 1$

⇒ $x^{2} - 5x -1 = 0$

We know that, 

$x = \frac{-b \± \sqrt{b^{2}- 4ac} }{2a}$

Here, a = 1 , b = (-5) , c = (-1)

Then,

⇒ $x = \frac{-(-5)\± \sqrt{(-5)^{2}-4(1)(-1)} }{2(1)} = \frac{5\± \sqrt{25 + 4 } }{2} = \frac{5\± \sqrt{29} }{2}$

Therefore.

$\frac{x^{2}-1}{x^{2}} = 1 - \frac{1}{x^{2}} = 1 - \frac{4}{(5\± \sqrt{29})^{2} }$

⇒  $1- \frac{4}{54+ 10\sqrt{29} } = \frac{50 + 10\sqrt{29} }{54+10 \sqrt{29} }$
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Hope my answer is helpful to you.