Question

If x= root 5+2 then the value of x power 2 +1/x power 2

Answer 1

Answer:

$Value\:of\:x^{2}+\frac{1}{x^{2}}=18$

Step-by-step explanation:

$Given \:x = \sqrt{5}+2 \:---(1)$

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

$x+\frac{1}{x}\\=\sqrt{5}+2+\sqrt{5}-2\\=2\sqrt{5}\:---(3)$

$Now,\\x^{2}+\frac{1}{x^{2}}\\=\left(x+\frac{1}{x}\right)^{2}-2\times x \times \frac{1}{x}\\=(2\sqrt{5})^{2}-2\\=4\times 5 - 2\\=20-2\\=18$

Therefore,

$Value\:of\:x^{2}+\frac{1}{x^{2}}=18$

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