Question

if x² + 1/x² = 27, find x² - 1/x²​

Answer 1

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GIVEN :–

• x² + 1/x² = 27

TO FIND :–

• Value of x² - 1/x² = ?

SOLUTION :–

• We know that –

➪ (a+b)² = a² + b² + 2ab

⇒ (x + 1/x)² = x² + 1/x² + 2(x)(1/x)

⇒ (x + 1/x)² = 27 + 2

⇒ (x + 1/x)² = 29

⇒ x + 1/x = √29 _____________eq.(1)

• We also know that –

➪ (a-b)² = a² + b² - 2ab

⇒ (x - 1/x)² = x² + 1/x² - 2(x)(1/x)

⇒ (x - 1/x)² = 27 - 2

⇒ (x - 1/x)² = 25

⇒ x - 1/x = √25

⇒ x - 1/x = ±5 _____________eq.(2)

• One more identity to use –

➪ (a-b)(a+b) = a² - b²

• Now multiply eq.(1) & eq.(2) –

⇒ (x + 1/x)(x - 1/x) = ±5(√29)

⇒ x² - 1/x² = ±5√29

▪︎ Hence , The Value of x² - 1/x² is ±5√29.