Find the value of the infinite nested radical expression:

$$
\sqrt{20 - \sqrt{20 - \sqrt{20 - \sqrt{20 - \ldots}}}}
$$

Question

Find the value of the infinite nested radical expression:

$$
\sqrt{20 - \sqrt{20 - \sqrt{20 - \sqrt{20 - \ldots}}}}
$$

Accepted Answer

**Solution:**

1. Let the value of the infinite nested radical be $x$:
   $$x = \sqrt{20 - \sqrt{20 - \sqrt{20 - \ldots}}}$$

2. Since the expression under the square root repeats infinitely, it is also equal to $x$:
   $$x = \sqrt{20 - x}$$

3. Square both sides to remove the square root:
   $$x^2 = 20 - x$$

4. Rearrange into standard quadratic form:
   $$x^2 + x - 20 = 0$$

5. Factor the quadratic:
   $$(x + 5)(x - 4) = 0$$

6. The solutions are:
   - $x = -5$
   - $x = 4$

7. Since the expression involves square roots, the value must be non-negative. Therefore, discard $x = -5$.

**Answer:**

The value of the expression is $\boxed{4}$.

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