Question

plz someone help me to do this no.​

Answer 1

Question :–

▪︎ If x² - 5x + 1 = 0 then find $\: { \bold{ {x}^{3} + \dfrac{1}{ {x}^{3}} = ? }} \:$

ANSWER :–

$\\ \: \implies { \bold{ {x}^{2} - 5x + 1 = 0 }} \:$

• We should write this as –

$\\ \: \implies { \bold{ {x}^{2} + 1 = 5x }} \:$

$\\ \: \implies { \bold{ \dfrac{{x}^{2} + 1}{x} = 5 }} \:$

$\\ \: \implies { \bold{x + \frac{1}{x} = 5 \: \: \: \: - - - - eq.(1)}}$

• Now take cube on both sides –

$\\ \: \implies { (\bold{x + \frac{1}{x} )^{3} = {(5)}^{3} }}$

• We know that –

$\\ \: { \huge { \star}} \: \: \: \large{ \green{ \boxed { \bold{{(a + b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b) }}}}$

• So that –

$\\ \: \implies { \bold{ {x}^{3} + \frac{1}{x^{3} } + 3( \cancel x)( \frac{1}{ \cancel x}) [x + \frac{1}{ x} ] = {(5)}^{3} }}$

$\\ \: \implies { \bold{ {x}^{3} + \frac{1}{x^{3} } + 3(x + \frac{1}{ x}) = {(5)}^{3} }}$

$\\ \: \implies { \bold{ {x}^{3} + \frac{1}{x^{3} } + 3(5) = 125 \: \: \: \: [using \: \: eq.(1)]}}$

$\\ \: \implies { \bold{ {x}^{3} + \frac{1}{x^{3} } + 15 = 125}}$

$\\ \: \implies { \bold{ {x}^{3} + \frac{1}{x^{3} } = 125 - 15}}$

$\\ \: \implies { \pink{ \boxed{ \bold{ {x}^{3} + \frac{1}{x^{3} } = 110}}}}$

$\rule{200}2$