Question

Simplify: ( a+ 1/a) 2 + ( a- 1/a) 2​

Answer 1

Given to simplify the value of :-

(a +1/a)² + (a -1/a)²

SOLUTION:-

Method - 1 :-

Expanding the 1st term using (a+b)² = a² + 2ab + b²

Expanding the 2nd term using (a-b)² = a²- 2ab + b²

= (a)² +(1/a)² + 2(a)(1/a) + (a)² + (1/a)² - 2(a)(1/a)

= a² + 1/a² + 2 + a² + 1/a² - 2

Keeping like term together

= a² + a² + 1/a² + 1/a² + 2 -2

= 2a² + 2/a²

So, the value of (a+1/a)² + (a -1/a)² is 2a² + 2/a²

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Method -2 :-

Without expanding the first term and second term we have algebraic identity that is

(a + b)² + (a -b)² is 2[a² + b²]

(a +1/a)² + (a -1/a)²

= 2[a² + 1/a²]

= 2a² + 2/a²

So, the value of (a+1/a)² + (a -1/a)² is 2a² + 2/a²

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Know more algebraic identities:-

( a + b )² + ( a - b)² = 2a² + 2b²

( a + b )² - ( a - b)² = 4ab

( a+b)(a -b ) = a² - b²

( 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)

a³ + b³ = (a +b)(a² - ab + b²)

a³ - b³ = (a-b)(a² + ab +b²)

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

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