Question

If $x+\frac{1}{x}=\sqrt{5}$, find the values of $x^2+\frac{1}{x^2}$ and $x^4+\frac{1}{x^4}$

Answer 1

$x+\frac{1}{x}=\sqrt{5}$

Square both sides,

${(x+\frac{1}{x})}^2=(\sqrt{5}{)}^2$

Apply $(a+b{)}^2=a^2+b^2+2ab$

$x^2+{\left(\frac{1}{x}\right)}^2+2\times x\times \frac{1}{x}=5$

$x^2+\frac{1}{x^2}+2=5$

$x^2+\frac{1}{x^2}=5-2=3$ .......... (1)

Again, Square both sides,

${(x^2+\frac{1}{x^2})}^2=3^2$

${\left(x^2\right)}^2+{\left(\frac{1}{x^2}\right)}^2+2\times x^2\times \frac{1}{x^2}=9$

$x^4+\frac{1}{x^4}+2=9$

$x^4+\frac{1}{x^4}=9-2=7$ ........ (2)