Question

Find the value of $ \sqrt {6 + \sqrt {6 + \sqrt {6 + \sqrt {6 + \ldots \ldots \ldots \infty } } } } $ =_______
A. $ 3 $
B. $ 2 $
C. $ 1 $
D. $ \pm 3 $

Answer 1

Hint: In the above question, first consider the whole expression to be equal to any variable (let it be x or n). then square the whole equation and compare with the given expression. Then we will be left with a quadratic equation, which can be solved by the factorization method. Hence, we got our solution.

Complete step by step solution:
For such questions, considering the whole term to be equal to any variable. Let the variable be x.
 $ x = $ $ \sqrt {6 + \sqrt {6 + \sqrt {6 + \sqrt {6 + \ldots \ldots \ldots \infty } } } } $ $ \ldots \left( 1 \right) $
Now, squaring both sides
 $ {x^2} = 6 + $ $ \sqrt {6 + \sqrt {6 + \sqrt {6 + \sqrt {6 + \ldots \ldots \ldots \infty } } } } $ $ \ldots \left( 2 \right) $
Now on observing the above mathematical expression we got to know that the equation $ \left( 1 \right) $ right-hand side is equal to the second term of equation $ \left( 2 \right) $
Hence, the value the value of equation $ \left( 1 \right) $ into equation $ \left( 2 \right) $
 $ {x^2} = 6 + x $
We are left with a quadratic equation, which we can solve easily by the factorization method. The equation is $ {x^2} - x - 6 = 0 $
Hence, splitting the middle term into two numbers which can be $ - 3,2 $ because their product gives the answer $ - 6 $ which is the constant term of the quadratic equation while their sum gives $ - 1 $ which is the coefficient of x.
Hence, splitting the middle term
 $ {x^2} - 3x + 2x - 6 = 0 $
Taking common from first two and last terms respectively,
 $ x(x - 3) + 2(x - 3) = 0 $
 $ \left( {x - 3} \right)\left( {x + 2} \right) = 0 $
Now, equating the two brackets to zero.
 $ x = 3, - 2 $
Hence, the solution of the above question is either $ 3 $ or $ - 2 $
So, the option which matches our answer is A.
So, the correct answer is “Option A”.

Note: Be careful while considering the variable because in some questions x is already taken. So, in those questions, consider the variable to be other than and x. And while finding the factors of the quadratic equation, Be careful about the signs of the numbers because wrong splitting will make the whole question wrong.