Question

if x=9-4√5 find x²-1/x²​

Answer 1

$\star\:\:\:\bf\large\underline\blue{Question:-}$

• If $x=9-4\sqrt{5}$, find the value of${x}^{2} - \frac{1}{ {x}^{2} }$

$\star\:\:\:\bf\large\underline\blue{Solution:-}$

$x = 9- 4 \sqrt{5}$

Therefore,

$\frac{1}{x}$

$= \frac{1}{9 - 4 \sqrt{5} }$

$= \frac{1}{9 - 4 \sqrt{5} } \times \frac{9 + 4 \sqrt{5} }{9 + 4 \sqrt{5} }$

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

$= \frac{9 + 4 \sqrt{5} }{81 - 80}$

$= 9+ 4 \sqrt{5}$

So,

$x + \frac{1}{x} = 9 - 4 \sqrt{5} + 9 + 4 \sqrt{5}$

$= 18$

Also,

$x - \frac{1}{x} = 9 - 4 \sqrt{5} - 9 - 4 \sqrt{5}$

$= - 8 \sqrt{5}$

So,

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

$= (x + \frac{1}{x} )(x - \frac{1}{x} )$

$= 18 \times ( - 8 \sqrt{5} )$

$= - 144 \sqrt{5}$

$\star\:\:\:\bf\large\underline\blue{Answer:-}$

• ${x}^{2} - \frac{1}{ {x}^{2} } = - 144 \sqrt{5}$

Answer 2

Step-by-step explanation:

$x=9-4\sqrt{5} \\\\ => \frac{1}{x} =\frac{1}{9-4\sqrt{5}} \\\\ rationalising \ factor=9+4\sqrt{5} \\\\ =>\frac{1}{9-4\sqrt{5}} \times \frac{9+4\sqrt{5}}{9+4\sqrt{5}} \\\\ => \frac{9+4\sqrt{5}}{(9-4\sqrt{5})(9+4\sqrt{5})} \\\\ => \frac{9+4\sqrt{5}}{9^2-(4\sqrt{5})^2} \\\\ => \frac{9+4\sqrt{5}}{81-16(5)}\\\\ =>\frac{9+4\sqrt{5}}{81-80} \\\\ =>9+4\sqrt{5}$

$\implies x^2-\frac{1}{x^2} \\\\ => (9-4\sqrt{5})^2-(9+4\sqrt{5})^2 \\\\ => 9^2+(4\sqrt{5})^2-2(9)(4\sqrt{5}) -[9^2+(4\sqrt{5})^2+2(9)(4\sqrt{5})] \\\\ =>81+80-72\sqrt{5}-81-80-72\sqrt{5} \\\\ => -72\sqrt{5}-72\sqrt{5}\\\\=>-144\sqrt{5}$