Question

if x = 4-√15, find the value of x +1/x​

Answer 1

Question :

If x = 4 - √15, find the value of x + 1/x​

Given :

x + 1/x = 4 - √15

To find :

Value of x + 1/x

Solution :

As we have, x = 4 - √15

$\therefore\sf \dfrac{1}{x} = \dfrac{1}{4 - \sqrt{15}} \\\\ \\ {\ \ \ \ \ }\rm{\star \ Rationalising \ the \ denominator : } \\\\ \\ : \implies\sf \dfrac{1}{x} = \dfrac{1(4 + \sqrt{15})}{(4 - \sqrt{15})(4 + \sqrt{15})} \\\\ \\ : \implies\sf \dfrac{1}{x} = \dfrac{4 + \sqrt{15}}{4^2 - (\sqrt{15})^2} \\\\ \\ : \implies\sf \dfrac{1}{x} = \dfrac{4 + \sqrt{15}}{16 - 15} \\\\ \\ : \implies\sf \dfrac{1}{x} = \dfrac{4 + \sqrt{15}}{1} \\\\ \\ : \implies\underline{\boxed{\sf{\dfrac{1}{x} = 4 + \sqrt{15} }}}$

Now value of x + 1/x :

$: \implies\sf x + \dfrac{1}{x} = 4 - \sqrt{15} + 4 + \sqrt{15} \\\\ \\ : \implies\underline{\boxed{\bold{x + \dfrac{1}{x} = 8 }}} \ \star$

Therefore,

Required value of x + 1/x = 8

Answer 2

$\underline{\underline{\bf{\maltese\:\:Question}}}$

$\sf{\bigstar\:\:\:If \:x\:=4-\sqrt{15}\:,\:Find\:\:the\:value\:\:of\:\:x\:+\:\dfrac{1}{x}}$

$\underline{\underline{\bf{\maltese\:\:Given}}}$

$\sf{\bullet\:\:\:x\:=4-\sqrt{15}}$

$\underline{\underline{\bf{\maltese\:\:To\:Find}}}$

$\sf{\bullet\:\:\:x+\dfrac{1}{x}}$

$\underline{\underline{\bf{\maltese\:\:Answer}}}$

$\sf{\bullet\:\:\:x\:+\:\dfrac{1}{x}=8}$

$\underline{\underline{\bf{\maltese\:\:Calculations}}}$

$\sf{Substitute\:\:the\:\:value\:\:of\:\:x\:\:(i,e.\:\:x\:=4-\sqrt{15})\:\: in \:\:x\:+\:\dfrac{1}{x}}$

$\sf{Then\:\:equation\:\:becomes}$

$\sf{x\:+\:\dfrac{1}{x}\:=\:4-\sqrt{15}+\dfrac{1}{4-\sqrt{15}}}$

$\bf{\:4-\sqrt{15}+\dfrac{1}{4-\sqrt{15}}}$

$\underline{\underline{\textsf{Multiply by the conjugate}\:\: \sf{\dfrac{4+\sqrt{15}}{4+\sqrt{15}}}}}$

$\sf{=4-\sqrt{15}+\dfrac{1\cdot \left(4+\sqrt{15}\right)}{\left(4-\sqrt{15}\right)\left(4+\sqrt{15}\right)}}$

$\sf{=4-\sqrt{15}+\dfrac{ \left(4+\sqrt{15}\right)}{\left(4-\sqrt{15}\right)\left(4+\sqrt{15}\right)}}$

$\sf{=4-\sqrt{15}+\dfrac{ \left(4+\sqrt{15}\right)}{4^2-\left(\sqrt{15}\right)^2}}$

$\sf{=4-\sqrt{15}+\dfrac{ \left(4+\sqrt{15}\right)}{16-15}}$

$\sf{=4-\sqrt{15}+\dfrac{ \left(4+\sqrt{15}\right)}{1}}$

$\sf{=4-\sqrt{15}+ \left(4+\sqrt{15}\right)}$

$\sf{=4-\sqrt{15}+ 4+\sqrt{15}\right}$

$\sf{=4+4}$

$\underline{\underline{\bf{=8}}}$