Question

Solution set of equation: \[x=\sqrt{12+\sqrt{12+\sqrt{12+.......\infty }}}\]
A) \[4\]
B) \[-4\]
C) \[3\]
D) \[-3\]

Answer 1

Hint: The infinite nested root can be compared to a variable. The variable can be used to replace the term inside the radical. We acquire the needed result by taking the square and solving the variable.

Complete step by step solution:
Here, we have an infinite number of nested square roots. We can equate the whole thing to x.
\[x=\sqrt{12+\sqrt{12+\sqrt{12+.......\infty }}}\]
As it never ending, the term inside the 1st square root can be written as \[12+x\]
\[x=\sqrt{12+x}\]

Now, take the square root on both sides. We get a quadratic equation as follows,
\[{{x}^{2}}=12+x\]
By rearranging the term we get:
\[{{x}^{2}}-x-12=0\]
Now, we have to factorize the above equation we get:
\[{{x}^{2}}-4x+3x-12=0\]
After simplifying the above equation we get:
\[x(x-4)+3(x-4)=0\]
Further solving the above equation we get:
\[(x-4)(x+3)=0\]
Therefore, x can have 2 values \[x=4,\,\,-3\]
But according to the question we have x as the square root of some numbers. As square roots cannot be negative. So, we take the value\[x=4\]
Therefore, the required solution is \[x=4\]. So, the correct option is option (A).

Note:
Infinite nested radicals is the name for this type of expression. This method can be used to solve any infinite nested radical. Despite the fact that this is an endless series, the sum is finite. It's important to remember that any integer contained within the square root radical is always positive. The value will become a complex number if the number inside the square root radical is negative. We can use any strategy to solve the quadratic equation because the solution will be unique.