Question

If x=5- 2 root 6 then find the value of x 2 +1/x 2

Answer 1

x=5-2√6
∴, x²=(5-2√6)²=25-2.5.2√6+24=49-20√6
1/x²=1/(5-2√6)²=1/(49-20√6)=(49+20√6)/(49-20√6)(49+20√6)=(49+20√6)/(2401-2400)=49+20√6
∴, x²+1/x²=49-20√6+49+20√6=98

Answer 2

Hey mate!

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Given :

$x = 5 - 2 \sqrt{6}$

To find :

$x {}^{2} + \frac{1}{x {}^{2} }$

Solution :

$x = 5 - 2 \sqrt{6} \\ \\ \frac{1}{x} = \frac{1}{5 - 2 \sqrt{6} } \times \frac{5 + 2 \sqrt{6} }{5 + 2 \sqrt{6} } \\ \\ \frac{1}{x} = \frac{5 + 2 \sqrt{6} }{(5) {}^{2} - (2 \sqrt{6} ) {}^{2} } \\ \\ \frac{1}{x} = \frac{5 + 2 \sqrt{6} }{25 - 24} \\ \\ \frac{1}{x} = 5 + 2 \sqrt{6}$

Now,

$x + \frac{1}{x} = 5 - \cancel{2 \sqrt{6}} + 5 + \cancel{ 2 \sqrt{6}} \\ \\ x + \frac{1}{x} = 5 + 5 \\ \\ x + \frac{1}{x} = 10$

Now, on squaring both sides ;

$(x + \frac{1}{x}) {}^{2} = (10) {}^{2} \\ \\ x {}^{2} + \frac{1}{x {}^{2} } + 2 = 100 \\ \\ x {}^{2} + \frac{1}{x {}^{2} } = 100 - 2 \\ \\ \boxed{ \bold{x {}^{2} + \frac{1}{x {}^{2} } = 98}}$

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Thanks for the question!

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