Answer:
We know that:
x = 3 + 2*√2
And we want to find the value of:
[tex]x - \frac{1}{x}[/tex]
First, we can rewrite the above expression as:
[tex]x - \frac{1}{x} = \frac{x^2}{x} - \frac{1}{x} = \frac{x^2 - 1}{x}[/tex]
Now we can replace the value of x by the one above, we will get:
[tex]\frac{x^2 - 1}{x} = \frac{(3 + 2*\sqrt{2})^2 - 1 }{3 + 2*\sqrt{2} } = \frac{3^2 + 2*2*\sqrt{2} + (2*\sqrt{2})^2 - 1 }{3 + 2*\sqrt{2} }[/tex]
Now we can simplify this to get:
[tex]\frac{9 + 4*\sqrt{2} + 4*2 - 1 }{3 + 2*\sqrt{2} } = \frac{16 + 4*\sqrt{2} }{3 + 2*\sqrt{2} }[/tex]
We could finally compute that last equation, to get:
[tex]\frac{9 + 4*\sqrt{2} + 4*2 - 1 }{3 + 2*\sqrt{2} } = \frac{16 + 4*\sqrt{2} }{3 + 2*\sqrt{2} } = 3.716[/tex]
