If a^2-5a+1=0 find the value of a+1/a and a^2+1/a^2
a2-5a+1=0
⟹ a2+1=5a
$\Rightarrow \frac{a^2 + 1}{a}$ = 5
$\Rightarrow \frac{a^2}{a} + \frac{1}{a}$ = 5
$\Rightarrow a + \frac{1}{a}$ = 5
$\therefore a + \frac{1}{a}$ = 5 [Ans]
$\therefore \left(a + \frac{1}{a}\right)^2$ = $5^2$
$\Rightarrow {\mathrm{a}}^2 + 2.{\mathrm{a}}\ldotp \frac{1}{{\mathrm{a}}} + \frac{1}{{\mathrm{a}}^2}$ = 25
$\Rightarrow {\mathrm{a}}^2 + 2 + \frac{1}{{\mathrm{a}}^2}$ = 25
$\Rightarrow {\mathrm{a}}^2 + \frac{1}{{\mathrm{a}}^2}$ = 25 – 2
$\Rightarrow {\mathrm{a}}^2 + \frac{1}{{\mathrm{a}}^2}$ = 23
$\therefore {\mathrm{a}}^2 + \frac{1}{{\mathrm{a}}^2}$ = 23 [Ans]