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Answer» $\Large{\underline{\underline{\mathfrak{\bf{Question}}}}}$ If ( x - 1/x ) = 3+2√2 , Find the VALUE of 1/4(x³ - 1/x³) $\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$ $\Large{\underline{\mathfrak{\bf{\pink{Given}}}}}$ ( x - 1/x ) = 3+2√2 .........(1) $\Large{\underline{\mathfrak{\bf{\orange{Find}}}}}$ Value of 1/4(x³ - 1/x³) $\Large{\underline{\underline{\mathfrak{\bf{Explanation}}}}}$ We know, ★ ( a-b)² = a² + b² - 2ab So, ★ ( x - 1/x )² = x² + 1/x² - 2.x.1/x ➥ (x - 1/x)² = x² + 1/x² - 2 Now, keep value by equ(1) ➥ (3+2√2)² = x²+1/x² - 2 ➥3²+(2√2)²+2.3.2√2 = x²+1/x² - 2 ➥ x² + 1/x² = (9+8+12√2)-2 ➥ x² + 1/x² = 15 + 12√2 Again, ★(a³-b³) = (a-b)(a²+b²+ab) So, ➥ (x³ - 1/x³ ) = (x - 1/x)(x² + 1/x² + x . 1/x) ➥(x³ - 1/x³ ) = (x - 1/x)(x² + 1/x² + 1) keep value by equ(1) and (2) ➥ (x³ - 1/x³ ) = ( 3+2√2)(15+12√2+1) ➥ (x³ - 1/x³ ) = (3+2√2)(16+12√2) ➥ (x³ - 1/x³ ) = 4(3+2√2)(4+3√2) ➥ (x³ - 1/x³ ) = 4(12+9√2+8√2+12) ➥ (x³ - 1/x³ ) = 4(24+17√2) Now, calculate , ➥ 1/4 (x³ - 1/x³ ) = 4(24+17√2)/4 ➥ 1/4 (x³ - 1/x³ ) = (24 + 17√2) (Ans.) _____________________

Answer»

$\Large{\underline{\underline{\mathfrak{\bf{Question}}}}}$

If ( x - 1/x ) = 3+2√2 , Find the VALUE of 1/4(x³ - 1/x³)

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\underline{\mathfrak{\bf{\pink{Given}}}}}$

$\Large{\underline{\mathfrak{\bf{\orange{Find}}}}}$

$\Large{\underline{\underline{\mathfrak{\bf{Explanation}}}}}$

We know,

★ ( a-b)² = a² + b² - 2ab

So,

★ ( x - 1/x )² = x² + 1/x² - 2.x.1/x

➥ (x - 1/x)² = x² + 1/x² - 2

Now, keep value by equ(1)

➥ (3+2√2)² = x²+1/x² - 2

➥3²+(2√2)²+2.3.2√2 = x²+1/x² - 2

➥ x² + 1/x² = (9+8+12√2)-2

➥ x² + 1/x² = 15 + 12√2

Again,

★(a³-b³) = (a-b)(a²+b²+ab)

So,

➥ (x³ - 1/x³ ) = (x - 1/x)(x² + 1/x² + x . 1/x)

➥(x³ - 1/x³ ) = (x - 1/x)(x² + 1/x² + 1)

keep value by equ(1) and (2)

➥ (x³ - 1/x³ ) = ( 3+2√2)(15+12√2+1)

➥ (x³ - 1/x³ ) = (3+2√2)(16+12√2)

➥ (x³ - 1/x³ ) = 4(3+2√2)(4+3√2)

➥ (x³ - 1/x³ ) = 4(12+9√2+8√2+12)

➥ (x³ - 1/x³ ) = 4(24+17√2)

Now, calculate ,

➥ 1/4 (x³ - 1/x³ ) = 4(24+17√2)/4

➥ 1/4 (x³ - 1/x³ ) = (24 + 17√2) (Ans.)

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