Question

x + 1/x =5 find x^2+1/x^2 and x^4+1/4

Answer 1

Given that,


x + 1/x = 5

To find the answer, we are going to use the identity, (a + b)² = a² + b² + 2ab


So,
$x + \frac{1}{x} = 5 \\ \\ = > (x + \frac{1}{x} ) {}^{2} = (5) {}^{2} \\ \\ = > {x}^{2} + \frac{1}{x}^{2} + 2 \times x \times \frac{1}{x} = 25 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 25 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 25 - 2 \\ \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 23$
So,

x² + 1/x² = 23

Again, squaring both sides,
${x}^{2} + \frac{1}{ {x}^{2} } = 23 \\ \\ = > ( {x}^{2} + \frac{1}{ {x}^{2} } ) {}^{2} = (23) {}^{2} \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } + 2 \times {x}^{2} \times \frac{1}{ {x}^{2} } = 529 \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 529 \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = 529 - 2 \\ \\ = > {x}^{4} + \frac{1}{ {x}^{4} } = 527$
so x⁴ + 1/x⁴ = 527


Hope it helps dear friend ☺️✌️