Question

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Answer 1

Question :

If x² - 3x + 1 = 0, then find the value of x³ - 1/x³.

Answer :

Given equation :

x² - 3x + 1 = 0

Let's find the value of ( x + 1/x ) from this equation

⇒ x² + 1 = 3x

Dividing every term with 'x'we get,

⇒ x + 1/x = 3

Now, let's find the value of ( x - 1/x )

We know that

( a - b )² = ( a + b )² - 4ab

Hence, ( x - 1/x )² = ( x + 1/x )² - 4( x )( 1/x )

⇒ ( x - 1/x )² = 3² - 4

⇒ ( x - 1/x )² = 9 - 4

⇒ ( x - 1/x )² = 5

It can be written as

⇒ ( x - 1/x )² = ( ± √5 )²

Taking square root on both sides

⇒ x - 1/x = ± √5

Cubing on both sides

⇒ ( x - 1/x )³ = ( ± √5 )³

Since ( a - b )³ = a³ - b³ - 3ab( a - b )

⇒ x³ - 1/x³ - 3( x )( 1/x )( x - 1/ x ) = ± 5√5

⇒ x³ - 1/x³ - 3( ± √5 ) = ± 5√5

⇒ x³ - 1/x³ ± 3√5 = ± 5√5

⇒ x³ - 1/x³ = ± 5√5 ± 3√5

⇒ x³ - 1/x³ = ± 8√5

Therefore the value of x³ - 1/x³ is ± 8√5.

Answer 2

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