Question

x=5-2√6 find the value of x^2+1/x^2​

Answer 1

Given :

• x = 5 - 2√6

To find :

• The value of x² + 1/x²

Solution :

~ The value of x is given to us, We are asked to find the value of x² + 1/x², Let's find the value of 1/x first, And then we'll rationalize it! Then we'll find x + 1/x by putting the values, And then we'll square it and find x² + 1/x²

According to the question,

⟶ x = 5 - 2√6

⟶ 1/x = 1/5 - 2√6

⟶ 1/x = 1/5 - 2√6 × 5 + 2√6/5 + 2√6

⟶ 1/x = 5 + 2√6/(5)² - (2√6)²

⟶ 1/x = 5 + 2√6/25 - 24

⟶ 1/x = 5 + 2√6

_

⟶ x + 1/x = 5 - 2√6 + 5 + 2√6

⟶ SOBS,

⟶ (x + 1/x)² = (5 - 2√6 + 5 + 2√6)²

⟶ x² + 1/x² = 5² - (2√6)² + 5² + (2√6)² - 2

⟶ 5² - (2√6)² + (2√6)² + 5² - 2

Cancelling -(2√6)² and (2√6)²,

⟶ 10² - 2

⟶ 100 - 2

⟶ 98

∴ Hence, The value of x² + 1/x² is 98.

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~ Additional information :

• ( a + b)² = a² + b² + 2ab

• ( a - b )² = a² + b² - 2ab

• ( a + b )² + ( a - b)² = 2a² + 2b²

• ( a + b )² - ( a - b)² = 4ab

• ( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

• a² + b² = ( a + b)² - 2ab

• (a + b )³ = a³ + b³ + 3ab ( a + b)

• ( a - b)³ = a³ - b³ - 3ab ( a - b)

• If a + b + c = 0 then a³ + b³ + c³ = 3abc