Solving the Infinite Nested Square Root Problem

We are asked to find the value of the expression: $$ x = \sqrt {12 + \sqrt {12 + \sqrt {12 + \ldots } } } $$ This is an infinite nested radical. We can solve this by setting up an equation.

Setting up the Equation

Let the value of the entire expression be denoted by '$x$'. Since the expression is infinite, we can see a pattern:

$$ x = \sqrt{12 + \underbrace{\sqrt{12 + \sqrt{12 + \ldots}}}_{x}} $$

Therefore, we can rewrite the expression as:

$$ x = \sqrt{12 + x} $$

Solving the Equation

To solve for '$x$', we first square both sides of the equation:

$$ x^2 = (\sqrt{12 + x})^2 $$ $$ x^2 = 12 + x $$

Now, we rearrange this into a standard quadratic equation ($ax^2 + bx + c = 0$):

$$ x^2 - x - 12 = 0 $$

Finding the Roots

We can solve this quadratic equation by factoring. We need two numbers that multiply to -12 and add up to -1. These numbers are -4 and 3.

So, we can factor the equation as:

$$ (x - 4)(x + 3) = 0 $$

This gives us two possible solutions for '$x$':

• $x - 4 = 0 \implies x = 4$
• $x + 3 = 0 \implies x = -3$

Determining the Correct Value

The original expression involves square roots. The square root symbol ($\sqrt{}$) conventionally denotes the principal (non-negative) root. Therefore, the value of the infinite nested square root must be positive.

Comparing our two solutions, '$x=4$' and '$x=-3$', we must choose the positive value.

Thus, the value of the expression is $4$.

Conclusion

The value of $\sqrt {12 + \sqrt {12 + \sqrt {12 + \ldots } } }$ is $4$. This corresponds to the option 4.000.