Question

if x+1/x=5,then find the value
${x }^{2} + \frac{1}{ {x}^{2} }$
​

Answer 1

Required Answer:-

Given:-

• $\sf{x + \frac{1}{x} = 5}$

To Find:-

• $\sf{ \bold{ {x}^{2} + \frac{1}{ {x}^{2} } }}$

♡ Solution:-

$\\\sf{\bold{x + \frac{1}{x} = 5}}$

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

$\\ \sf{ \implies{ {(x)}^{2} + 2 \times x \times \frac{1}{x} + ( \frac{1}{x} ) {}^{2} = 25}}$

$\\ \sf{ \implies{ {x}^{2} + 2 + \frac{1}{ {x}^{2} } = 25}}$

$\\ \sf{ \implies{ {x}^{2} + \frac{1}{ {x}^{2} } = 25 - 2}}$

$\\ \sf{ \therefore{ {x}^{2} + \frac{1}{ {x}^{2} } = \orange{23}}}$

Another way to solve:-

$\\\sf{\bold{ {x}^{2} + \frac{1}{ {x}^{2} }} }$

$\\ \sf{ \implies{(x) {}^{2} + ( \frac{1}{x} ) {}^{2} }}$

$\\ \sf{ \implies{(x + \frac{1}{x} ) {}^{2} - 2 \times x \times \frac{1}{x} }}$

$\\ \sf{ \implies{(5) {}^{2} - 2}}$

$\\ \sf{ \implies{25 - 2}}$

$\\ \sf{ \therefore{x {}^{2} - \frac{1}{ {x}^{2} } = \orange{23}}}$

$\sf{ \underline{ \underline{ \purple{More \: important \: formulas:-}}}}$

$\sf{ \bold{(a + b) {}^{2} } = {a}^{2} + 2ab + {b}^{2} } \\ \\ \sf{ \bold{(a - b) {}^{2} } = {a}^{2} - 2ab + {b}^{2} } \\ \\ \sf{ \bold{{a}^{2} - {b}^{2} } = (a + b)(a - b)} \\ \\ \sf{ \bold{ {a}^{2} + {b}^{2}} = (a + b) {}^{2} - 2ab} \\ \\ \sf{ \bold{ {a}^{2} + {b}^{2}} = (a - b) {}^{2} + 2ab}$