Question

if. a-1/a=4 ,then a^2+1/a^2=​

Answer 1

$\large \sf\underline{ \underline{ \red{given: }}} \\ \\ \sf{a - \frac{1}{a} = 4 }$

$\large \sf\underline{ \underline{ \red{to \: find : }}} \\ \\ \sf{ {a}^{2} + \frac{1}{ {a}^{2} } }$

$\large \sf\underline{ \underline{ \red{to \: know : }}} \\ \\ \boxed{ \tt{(x + {y)}^{2} = {x}^{2} + {y}^{2} + 2xy }}$

$\large \sf\underline{ \underline{ \red{solution : }}}$

$\\ \\ \sf{a - \frac{1}{a} = 4} \\ \\ \\ \sf{squaring \: both \: sides ..} \\ \\ \\ \sf{(a - { \frac{1}{a} )}^{2} } = {4}^{2}$

According to identity ,

Here ,

• x = a

• y = 1/a

Putting values ,

$\\ \sf{ {a}^{2} + { (\frac{1}{a} )}^{2} + 2( \cancel{a})( \frac{1}{ \cancel{a}} ) = 16 } \\ \\ \\ \sf{ {a}^{2} + \frac{1}{ {a}^{2} } + 2 = 16 } \\ \\ \\ \implies \boxed{\sf{ \pink {{a}^{2} + \frac{1}{ {a}^{2} } = 14 }}}$

Other identities :-

• ( x + y )² = x² + 2xy + y²

• ( x + y ) ( x - y ) = x² - y²

• ( x + a ) ( x + b ) = x² + ( a + b )x + ab