Question

If $\bf{x = 5 - {5}^{ \frac{2}{3} } - {5}^{ \frac{1}{3} }}$. Then find the Value of (x³ - 15x² + 60x - 15) = ?

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Answer 1

AnswEr :

$\longrightarrow \large\sf{x = 5 - {5}^{ \frac{2}{3} } - {5}^{ \frac{1}{3} }}$

• Interchanging Signs

$\longrightarrow \large\sf{ - x = - 5 + {5}^{ \frac{2}{3} } + {5}^{ \frac{1}{3} }}$

$\longrightarrow \large\sf{ 5- x = {5}^{ \frac{2}{3} } + {5}^{ \frac{1}{3} }}$

• Cubing Both Sides

$\longrightarrow \large\sf{ (5- x)^{3} = ( {5}^{ \frac{2}{3} } + {5}^{ \frac{1}{3} })^{3} }$

• Solving LHS Side :

$\longrightarrow \sf{(5)}^{3} - {(x)}^{3} - 3 \times 5 \times x(5 - x)$

$\longrightarrow \sf125 - {x}^{3} - 15x(5 - x)$

$\longrightarrow \sf{125 - {x}^{3} - 75x + 15 {x}^{2}}$

• Solving RHS Side :

$\longrightarrow \sf(5^{\frac{2}{ \cancel3}})^{ \cancel3} + (5^{\frac{1}{ \cancel3}})^{ \cancel3} + 3 \times {5}^{ \frac{2}{3} } \times {5}^{{\frac{1}{3} }} ( {5}^{ \frac{2}{3} } + {5}^{ \frac{1}{3} })$

$\longrightarrow \sf( {5})^{2} + ( {5})^{1} + 3 \times {5}^{ (\frac{2}{3} + \frac{1}{3}) } ( {5}^{ \frac{2}{3} } + {5}^{ \frac{1}{3} })$

• we can change $\sf{({5}^{ \frac{2}{3} } + {5}^{ \frac{1}{3}}) =( 5 - x) }$

$\longrightarrow \sf{25 + 5 + (3 \times {5}^{ \cancel\frac{3}{3}}) ( 5 - x)}$

$\longrightarrow \sf 30 + 15 ( 5 - x)$

$\longrightarrow \sf30 + 75 - 15x$

$\longrightarrow \sf105 - 15x$

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• Combining LHS and RHS Together :

$\longrightarrow \sf{125 - {x}^{3} - 75x + 15 {x}^{2}= 105 - 15x}$

$\longrightarrow \sf{125 - 105 - {x}^{3} - 75x + 15x+ 15 {x}^{2}= 0}$

$\longrightarrow \sf{20- {x}^{3} - 60x+ 15 {x}^{2}= 0}$

$\longrightarrow \sf{20 = {x}^{3} + 60x - 15 {x}^{2}}$

$\longrightarrow \sf{15 + 5= {x}^{3} + 60x - 15 {x}^{2}}$

$\longrightarrow \sf{x}^{3} - 15 {x}^{2} + 60x - 15 = 5$

⠀

$\therefore \underline{\sf{Value \: of \: {x}^{3} -15{x}^{2} + 60x - 15 = 5}}$