Question

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Answer 1

Gɪᴠᴇɴ :-

• x + 1/x = 5

Tᴏ Fɪɴᴅ :-

• x³ - 1/x³ = ?

Fᴏʀᴍᴜʟᴀ ᴜsᴇᴅ :-

• (a + b)² = a² + b² + 2ab
• a² + b² - 2ab = (a - b)²
• (a - b)³ = a³ - b³ - 3ab(a - b)

Sᴏʟᴜᴛɪᴏɴ :-

➪ (x + 1/x) = 5

Squaring Both sides, we get,

➪ (x + 1/x)² = (5)²

➪ x² + 1/x² + 2 * x * 1/x = 25

➪ (x² + 1/x²) + 2 = 25

➪ (x² + 1/x²) = 25 - 2

➪ (x² + 1/x²) = 23

Subtracting 2 From Both sides Now,

➪ (x² + 1/x²) - 2 = 23 - 2

➪ (x² + 1/x²) - 2 * x * 1/x = 21

➪ (x - 1/x)² = 21

Square - Root Both sides Now,

➪ (x - 1/x) = √21 ------------ Eqn(1)

Cube Both sides Now,

➪ (x - 1/x)³ = (√21)³

➪ x³ - 1/x³ - 3 * x * 1/x * (x - 1/x) = 21√21

➪ (x³ - 1/x³) - 3(x - 1/x) = 21√21

Putting Value from Eqn(1) Now,

➪ (x³ - 1/x³) - 3 * √21 = 21√21

➪ (x³ - 1/x³) = 21√21 + 3√21

➪ (x³ - 1/x³) = 24√21 (Option A) (Ans.)

Answer 2

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

$\star \: {\tt{x + \frac{1}{x} = 5 }}$

${\bf{\blue{\underline{To find:}}}}$

$\star \: {\tt{ {x}^{3} - \frac{1}{ {x}^{3} } }}$

${\bf{\blue{\underline{Now:}}}}$

Take,

$\implies\: {\tt{x + \frac{1}{x} = 5 }}$

Squaring both side,

$\implies\: {\tt{ \big(x + \frac{1}{x} \big) ^{2} = ( {5}^{2} ) }}$

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

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

$\implies\: {\tt{ {x}^{2} + \frac{1}{ {x}^{2} } = 23}}$

Now it can be written as

$\implies\: {\tt{ {x}^{2} + \frac{1}{ {x}^{2} } - 2 \times x \times \frac{1}{x} = 23 - 2}}$

$\implies\: {\tt{ \bigg( {x - \frac{1}{x} } \bigg) ^{2} = 21}}$

$\implies {\boxed {\tt{ \purple{ \: x - \frac{1}{x} = \sqrt{21} }}}}$

Cubing both side,

$\implies\: {\tt{ \big(x - \frac{1}{x} \big) ^{3} = ( \sqrt{21} ) ^{3} }}$

$\implies\: {\tt{ {x}^{3} - 3 \times {x}^{2} \times \frac{1}{x} + 3 \times x \times \frac{1}{ {x}^{2} } - \frac{1}{ {x}^{3} } = ( \sqrt{21} ) ^{3} }}$

$\implies\: {\tt{ {x}^{3} - 3x+ 3 \times \frac{1}{ {x} } - \frac{1}{ {x}^{3} } = ( \sqrt{21} ) ^{3} }}$

$\implies\: {\tt{ {x}^{3} - \frac{1}{ {x}^{3} } - 3 \big(x - \frac{1}{ {x} } \big ) = ( \sqrt{21} ) ^{3} }}$

We know that,

${\boxed {\tt{ \purple{ \: x - \frac{1}{ x } = \sqrt{21} }}}}$

$\implies\: {\tt{ {x}^{3} - \frac{1}{ {x}^{3} } - 3( \sqrt{21} ) = ( { \sqrt{21} )}^{3} }}$

$\implies\: {\tt{ {x}^{3} - \frac{1}{ {x}^{3} } =( { \sqrt{21} )}^{3} + 3( \sqrt{21} )}}$

$\implies\: {\tt{ {x}^{3} - \frac{1}{ {x}^{3} } = \sqrt{21} \times( \sqrt{21} \times \sqrt{21} ) + 3( \sqrt{21} )}}$

$\implies\: {\tt{ {x}^{3} - \frac{1}{ {x}^{3} } =21 \sqrt{21} + 3( \sqrt{21} )}}$

${\boxed {\tt{ \purple{ \: {x}^{3} - \frac{1}{ {x}^{3} } = 24\sqrt{21} }}}}$