Question

if a+1/a=2,a^4+1/a^2​

Answer 1

$\huge\mathfrak{Answer}$

2

$\rule{200}2$

$\huge\textsf{Explanation}$

Given

$\sf a + \frac{1}{a} = 2$

To find

$\sf a^4 + \frac{1}{a^2}$

Now, square both the sides :-

$\sf(a + \frac{1}{a})^2 = (2)^2 \\\\\implies \ a^2 + \frac{1}{a^2} + 2 = 4 \\\\ (as, (a + b)^2 = a^2 + 2ab + b^2)\\\\\implies a^2 + \frac{1}{a^2} = 4 -2 \\\\\implies a^2 + \frac{1}{a^2} = 2 \\\\Now, subtract \ 2 \ from \ both \ sides\\\\\implies a^2 + \frac{1}{a^2} - 2 = 2 - 2 \\\\\implies (a - \frac{1}{a})^2 = 0 \\\\\implies a - \frac{1}{a} = 0$

$\sf and\ we\ know \ that \\\\a + \frac{1}{a} = 2 \\\\so, add\ the\ two\ equations :- \\\\a + \frac{1}{a} + a - \frac{1}{a}= 2 + 0 \\\\\implies 2a = 2 \\\\\implies a = 1$

Now, we got the value of a, so we can find the value of :- $\sf a^4 + \frac{1}{a^2} \\\\\implies 1^4 + \frac{1}{1^2}\\\\\implies 1 + 1 = 2 \\\\Hence, a^4 + \frac{1}{a^2} = 2$

$\rule{200}2$