Question

if (a²+1/a²)=14 then find (a²-1/a²)​

Answer 1

Answer:

a²+1/a²=14

or,a²+1=14a²

or,14a²-a²=1

or,13a²=1

or,a²=1/13

So, a²-1/a²=(1/13)²-1/(1/13)²

=(1/169-1)/(1/169)

={(1-169)/169}(1/169)

= -168/169×169

= -168

Answer 2

       $\underline{\bold{Answer:}}$

       $8\sqrt3$

       $\underline{\bold{Step-by-step-explaination:}}$

       $\text{It is given that:}$

       $\boxed{a^{2} + \frac{1}{a^{2}} = 14}$

       $\text{Squaring both sides}$

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

       $\text{We know that }(a+b)^{2} = a^{2} +b^{2} +2ab$

$\implies \boxed{(a^{2})^{2} + (\frac{1}{a^{2}})^{2} + 2(a^{2})(\frac{1}{a^{2}}) = 196 }$

$\implies \boxed{a^{4} + \frac{1}{a^{4}} + 2 = 196}$

$\implies \boxed{a^{4} + \frac{1}{a^{4}} = 194} \text{ ------(i)}$

       $\text{We know that }(a-b)^{2} = a^{2} +b^{2} -2ab$

$\implies \boxed{(a^{2}- \frac{1}{a^{2}})^{2} = (a^{2})^{2} +(\frac{1}{a^{2}})^{2} - 2(a^{2})(\frac{1}{a^{2}}) }$

$\implies \boxed{(a^{2}- \frac{1}{a^{2}})^{2} = a^{4} +\frac{1}{a^{4}} - 2}$

       $\text{From (i)}$

$\implies \boxed{(a^{2}- \frac{1}{a^{2}})^{2} = 194 - 2}$

$\implies \boxed{(a^{2}- \frac{1}{a^{2}})^{2} = 192}$

$\implies \boxed{a^{2}- \frac{1}{a^{2}} = \sqrt{192}}$

$\implies \boxed{a^{2}- \frac{1}{a^{2}} = 8\sqrt3}$