Question

a=4-√15 then a^2+1/a^2=​

Answer 1

$\huge\rm\color{PALEVIOLETRED} Solution$

$\rm{a = 4 - \sqrt{15} }$

$\rm{ \frac{1}{a} = \frac{1}{4 - \sqrt{15} } }$

rationalise the denominator.

$\rm{ \frac{1}{a} = \frac{1}{4 - \sqrt{15} } \times \frac{4 + \sqrt{15} }{4 + \sqrt{15} } }$

$\rm{ \frac{1}{a} = \frac{4 + \sqrt{15} }{ {4}^{2} - {(\sqrt{15})}^{2} } }$

$\rm{ \frac{1}{a} = \frac{4 + \sqrt{15} }{ 16 - 15}}$

$\rm{ \frac{1}{a} = {4 + \sqrt{15} }}$

$\rm{a + \frac{1}{a} = 4 - \sqrt{15} + 4 + \sqrt{15} }$

$\rm{a + \frac{1}{a} = 8}$

Squaring on both sides

$\rm{{(a + \frac{1}{a})}^{2} = {8}^{2} }$

$\rm{{ {a}^{2} + \frac{1}{{a}^{2}} + 2 }= 64 }$

$\rm{{ {a}^{2} + \frac{1}{{a}^{2}} }= 64 - 2}$

$\fbox{\rm{{ {a}^{2} + \frac{1}{{a}^{2}} }= 62}}$