Question

If (x + 1/x) = 3, then find the value of: a) (x² + 1/x²) and b) (x⁴ + 1/x⁴).​

Answer 1

Question:

If $x+\frac{1}{x}=3$ .Then find

• $(x^2 + \frac{1}{x^2})$

• $(x^4 + \frac{1}{x^4})$

Given:

$x+\frac{1}{x}=3$

To Find:

• $(x^2 + \frac{1}{x^2})$

• $(x^4 + \frac{1}{x^4})$

Solution:

$x+\frac{1}{x}=3$

$\textsf{squaring on both sides}$

$(x+\frac{1}{x})=3^2\\\\x^2+\frac{1}{x^2}=3\times3\\\\x^2+\frac{1}{x^2}+2=9\\\\x^2+\frac{1}{x^2}=9-2\\\\x^2+\frac{1}{x^2}=7\\\\\boxed{x^2+\frac{1}{x^2}=7}$

$x^2+\frac{1}{x^2}=7}$

$\textsf{squaring on both sides}$

$(x^2+\frac{1}{x^2})^2=7^2}\\\\((x^2)^2+(\frac{1}{x^2})^2)+2=49\\\\x^4+\frac{1}{x^4}=49-2\\\\x^4+\frac{1}{x^4}=47\\\\\boxed{x^4+\frac{1}{x^4}=47}$

Answers:

$\boxed{x^2+\frac{1}{x^2}=7}$

$\boxed{x^4+\frac{1}{x^4}=47}$