Question

UPSC square + 1 upon x square is equal to 51 then x cube minus one upon x cube is equal to what

Answer 1

x² + 1/x² = 27

x² + 1/x² - 2 = 27 - 2

x² + 1/x² - 2 × x × 1/x = 25

( x - 1/x )² = 5²

x - 1/x = ± 5

Answer 2

$\Large{\bold{Hey\:\;\;Mate\;\;\:Here\:\:\:Is\:\;\;Your\;\;\:Answers}}$

$\underline{\bold{Given}}$

$\bold{{x}^{2} + \frac{1}{ {x}^{2} } = 51}$

$\bold{And\: We \:Need \:To\: Find...}$

$\bold{{x}^{3} - \frac{1}{ {x}^{3} } = ?}$

So Now Let's Move For Solution ....

$\underline{\bold{Solution}}$

Now According To Question It's Given That ${x}^{2} + \frac{1}{ {X}^{2} } = 51$

Now To Find ...

$\bold{ {x}^{3} - \frac{1}{ {x}^{3} }}$

Follow The Simple Step ...

$\underline{\bold{Step-1)\: Find\: Value \:Of\:\:\:{x}\:- \frac{1}{ x} }}$

So Now We Know By Identity ...

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

That

$\bold{ {(x\: - \:\frac{1}{x})}^{2} = \: {x}^{2} \:+ \:\frac{1}{ {x}^{2} } - 2(x) (\frac{1}{x} )}$

That Is ...

$\bold{{(x - \frac{1}{x})}^{2} = {x}^{2} + \frac{1}{ {x}^{2} } \: - 2 \: (by \: \: \: cancelling \: x)}$

So Now In Question Its Given That

$\bold{ {x}^{2} + \frac{1}{ {x}^{2} } = 51}$

Now Putting This Value In the Obtained Equation We Have ...

$\bold{ {(x - \frac{1}{x}) }^{2} = 51 - 2}$

That Is ...

$\bold{ {(x - \frac{1}{x} )}^{2} = 49}$

Now Moving Square To RHS We Have

$\bold{x - \frac{1}{x} = \sqrt{49}}$

That Is ...

$\bold{x - \frac{1}{x} = 7}$

So Now We Have Value Of

$\boxed{\bold{x - \frac{1}{x} = 7}}$

$\undelrine{\bold{Step -2)\:Solve\:For\:{x}^{3} -\frac {1}{{x}^{3}} }$

Now By Identity ..

$\boxed{ \bold{{(a - b)}^{3} = {a}^{3} - {b}^{3} - 3ab \: (a - b)}}$

We Have ..

$\bold{ {(x - \frac{1}{x})}^{3} = {x}^{3} - \frac{1}{ {x}^{3} } - 3(x)( \frac{1}{x}) \: (x - \frac{1}{x} )}$

That is ...

$\bold{{(x - \frac{1}{x} )}^{3} = {x}^{3} - \frac{1}{ {x}^{3} } - 3(x - \frac{1}{x} )}$

Now In Above Method We Have Found Value Of

$\bold{x - \frac{1}{x} = 7}$

Now Putting This Value In This Obtained Equation We Have ...

$\bold{ {7}^{3} = {x}^{3} - \frac{1}{ {x}^{3} } - 3(7)}$

That Is ...

$\bold{343 = {x}^{3} - \frac{1}{ {x}^{3} } - 21}$

That is ...

$\bold{343 + 21 = {x}^{3} - \frac{1}{ {x}^{3} } }$

That Is ..

$\bold{364 = {x}^{3} - \frac{1}{ {x}^{3} } }$

Therefore Value Of

$\boxed{\bold{ {x}^{3} - \frac{1}{ {x}^{3} } = 364}}$

$\bold{Hence\:The\: Required\:Answer\:Is....}$

$\boxed{\mathfrak{\bold{364} } }$

$\Large{\bold{Thanks...}}$