Question

what is the value of
$\sqrt{6 + \sqrt{6 + \sqrt{6} } }$

Answer 1

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Answer 2

SOLUTION :  

Option (b) is correct :  3

Let x = [√6 + √6 + √6 + √6 + …..]

x = √(6 + x)...........(1)

On squaring both sides,

x² = 6 + x  

x² - x - 6 = 0

x² - 3x + 2x - 6 = 0

[By middle term splitting]

x(x - 3) + 2(x - 3) = 0

(x - 3) (x + 2 ) = 0

(x - 3) = 0 (x + 2 ) = 0

x = 3 or x = - 2  

On putting x = 3 in eq 1,

x = √(6 + x)

3 = √(6 +3)

3 = √9 = 3  

This is possible  

On putting x = - 2 in eq 1,

x = √(6 + x)

-2 = √(6 - 2)

- 2 = √4 = 2

This is not possible .

Hence , the value is 3 .

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