If-x2-1-x2-18-find the values of-x-1-x-and-x-1-x-

X^2+1/x^2=18 Adding 2 to both sides, x^2+2+1/x^2=18+2 ⇒ (x+1/x )^2=20 ∴ x+1/x=√(20) Again x^2+1/x^2=18 Subtracting 2 from both sides, x^2 2+1/x^2=18 2=16 (x ...

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Standard VIII

Mathematics

Algebraic Identities

If x2+1/x2=18...

Question

$I f x^{2} + \frac{1}{x^{2}} = 18, find the values of x + \frac{1}{x} a n d x - \frac{1}{x} .$

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Solution

$x^{2} + \frac{1}{x^{2}} = 18 Adding 2 to both sides, x^{2} + 2 + \frac{1}{x^{2}} = 18 + 2 \Rightarrow {(x + \frac{1}{x})}^{2} = 20 ∴ x + \frac{1}{x} = \sqrt{20} A g a i n x^{2} + \frac{1}{x^{2}} = 18 Subtracting 2 from both sides, x^{2} - 2 + \frac{1}{x^{2}} = 18 - 2 = 16 {(x - \frac{1}{x})}^{2} = (4)^{2} ∴ x - \frac{1}{x} = 4 H e n c e x + \frac{1}{x} = \sqrt{20} a n d x - \frac{1}{x} = 4$

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If x2+1x2=18, find the values of x+1x and x−1x.

x2+1x2=18Adding 2 to both sides,x2+2+1x2=18+2⇒ (x+1x)2=20∴ x+1x=√20Again x2+1x2=18Subtracting 2 from both sides,x2−2+1x2=18−2=16(x−1x)2=(4)2∴ x−1x=4Hence x+1x=√20 and x−1x=4

$I f x^{2} + \frac{1}{x^{2}} = 18, find the values of x + \frac{1}{x} a n d x - \frac{1}{x} .$