Question

X+1/x=3,find x^2- 1/x^2

Answer 1

Hi dear,

METHOD 1,

x +1/x = 3

Square on both sides,

(x +1/x)² = 3²

x² +1/x² +2(x×1/x) =9

x² +1/x² +2 = 9

x² +1/x² =9-2

x² +1/x² = 7

Then, add -2(x×1/x) on both sides,

x² +1/x² -2(x×1/x) = 7-2(x×1/x)

(x-1/x)² = 7-2

(x-1/x)² = 5

x -1/x = √5

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Now,

x² -1/x² = (x+1/x) (x-1/x)

x² -1/x² = (3)(√5)

x² -1/x² = 3√5

__________________

METHOD 2,

x +1/x = 3

Square on both sides,

(x+1/x)² = 3²

x² +1/x² +2(x×1/x) = 9

x² +1/x² +2 =9

x² +1/x² =9-2

x² +1/x² =7

Again square on both sides,

(x² +1/x²)² = 7²

x⁴ +1/x⁴ +2(x² × 1/x²) =49

x⁴ +1/x⁴ +2=49

x⁴ +1/x⁴ =49-2

x⁴ +1/x⁴ =47

Add -2(x² × 1/x²) on both sides,

x⁴ +1/x⁴ -2(x² × 1/x²) =47 -2(x² × 1/x²)

(x² -1/x²)² = 47-2

x² -1/x² = √45

x² -1/x² = √(9×5)= 3√5



I hope this will help you

-by ABHAY