Another nested radical?

Let $\quad x = \sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+...}}}}$ .

Then $\quad x = \sqrt{6+x}$

$\quad \Rightarrow x^2=6+x \quad \Rightarrow x^2-x-6 = 0 \quad \Rightarrow (x-3)(x+2) = 0 \quad \Rightarrow x = 3$

Therefore, $\quad 6 + \sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6+...}}}} = 6 + x = 6 + 3 = \boxed{9}$

Lets isolate first so let start with $6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6...}}}}=y$ - substract $6$ then expression becomes $y-6=\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6...}}}}$ square both sides so it is, $(y-6)^2=6+\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{6...}}}}$ ] clearly the nested radical is $y$ so the expression is $y^2-12y+36=y$ $y^2-13y+36=0$ then clearly the solution is $y=9,4$ and from the choices we are give $9$ as the answer. Note 4 is not a solution because the nested radical cannot it negative.

6 + any random positive number equals >6, and in the options, only 9 is higher than 6

Obviously the answer have to be greater than 6!

This problem can be solved much easily by just analysing the options. We know that √6=2.236(approx.) √[6+√(6+2.236)]=√[ 6 +√8.23] If we keep going, we will come nearer to 9. Nearest option will be 9.

Let x = 6 + √(6 + √(6 + √(6 + √(6... (infinite polynomial)

Solution:

1. Inspect and split parts of polynomial

x = 6 + √[ (6 + √(6 + √(6 + √(6...]

1. The bracketed part is still an infinite polynomial which is equal to the original expression "x"

2. Therefore we can now rewrite the original expression as

x = 6 + √x

1. Solve for x

            x - 6 = √x (subtract 6 to both sides)
   
           (x - 6)²  = (√x)²  (squaring both sides)
   
            x²  - 12x + 36 = x (simplify)
   
            x² - 13x + 36 = 0 (subtract x to both sides)
   
            (x - 4)(x - 9) = 0 (getting roots)
   
            x = 4 (null), x = 9 (acceptable answer)
   

No need to calculate. If not multiple choice question, then

Let x = 6 + Sqrt x

x^2 - 13 x + 36 = 0

(x - 4)(x - 9) = 0

x = 9 as x > 6.

Let the given expression is equal to x Then x = 6 + square root of x (x - 6)^2 = x x^2 - 13 x + 36 = 0 ( x - 4 )( x - 9 ) = 0 x = 9 x = 4 ................(refused)

The Answer given by Mardokay Mosazghi is very right theoretically but by problem solving approach and after analyzing the options the answer should be more than 6 Thus leaving us to the only answer 9 !!

6 + any positive quantity must be greater than 6 :D

It was so easy. Please post a good problem....