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ANEK [815]
3 years ago
13

If x=2+√5 find the value of x²-1/x²

Mathematics
2 answers:
e-lub [12.9K]3 years ago
7 0

Answer:

8\sqrt{5}

Step-by-step explanation:

x = 2 + \sqrt{5}\\\\ x^{2} = (2+ \sqrt{5})^{2} \\\\ \ \ \ \ = 2^{2}+2* \sqrt{5}*2+( \sqrt{5})^{2}\\\\

 = 4 + 4 \sqrt{5}+5\\\\= 9+4 \sqrt{5}

\frac{1}{x^{2}}=\frac{1}{9+4\sqrt{5}}\\\\=\frac{1*(9-4\sqrt{5}}{(9+4\sqrt{5})(9-4\sqrt{5})}\\\\=\frac{9-4\sqrt{5}}{9^{2}-(4\sqrt{5})^{2}}\\\\=\frac{9-4\sqrt{5}}{81-4^{2}(\sqrt{5})^{2}}\\\\=\frac{9-4\sqrt{5}}{81-16*5}\\\\=\frac{9-4\sqrt{5}}{81-80}\\\\=\frac{9-4\sqrt{5}}{1}\\\\=9-4\sqrt{5}

x^{2}-\frac{1}{x^{2}}= 9 + 4\sqrt{5} -(9 - 4\sqrt{5})\\\\

            = 9 + 4\sqrt{5} - 9 + 4\sqrt{5}\\\\= 9 - 9 + 4\sqrt{5} + 4\sqrt{5}\\\\= 8\sqrt{5}

natta225 [31]3 years ago
3 0

Answer:

{ \tt{ {x}^{2}  -  \frac{1}{ {x}^{2} } }} \\  = { \tt{ {(2 +  \sqrt{5} )}^{2} -  \frac{1}{ {(2 +  \sqrt{5}) }^{2} }  }} \\  = { \tt{ \frac{(2 +  \sqrt{5} ) {}^{4}  - 1}{ {(2 +  \sqrt{5} )}^{2}  } }} \\  = { \tt{ \frac{(9 + 4 \sqrt{5}) {}^{2}  }{ {(9 +  4\sqrt{5}) }}}} \\  = { \tt{9 + 4 \sqrt{5} }}

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