https://r2.erweima.ai/mathbot/image/1722088439284xh53vvwp.png

The expression provided is:

\[ \sqrt{15 - \sqrt{15 - \sqrt{15 - \ldots}}} \]

To find the value of this infinite nested radical, let's denote it by \( x \). Therefore, we can write:

\[ x = \sqrt{15 - x} \]

To eliminate the square root, square both sides of the equation:

\[ x^2 = 15 - x \]

Rearrange this equation into a standard quadratic form:

\[ x^2 + x - 15 = 0 \]

To solve this quadratic equation, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -15 \):

\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-15)}}{2 \cdot 1} \]
\[ x = \frac{-1 \pm \sqrt{1 + 60}}{2} \]
\[ x = \frac{-1 \pm \sqrt{61}}{2} \]

Since \( x \) represents a length (and thus must be non-negative), we take the positive root:

\[ x = \frac{-1 + \sqrt{61}}{2} \]

Therefore, the value of the infinite nested radical expression is:

\[ \boxed{\frac{-1 + \sqrt{61}}{2}} \]

Would you like more details on the steps, or do you have any questions?

Here are five related questions you might be interested in:
1. How do you solve other types of infinite nested radicals?
2. What are the properties of quadratic equations?
3. How do you solve quadratic equations by factoring?
4. How do you derive the quadratic formula?
5. What are the applications of nested radicals in real life?

**Tip:** When dealing with infinite nested radicals, setting up an equation for the expression can simplify the problem significantly.

Explore how to solve the infinite nested radical \( \sqrt{15 - \sqrt{15 - \sqrt{15 - \ldots}}} \) step-by-step. Understand the concept of infinite nested radicals, quadratic equations, and square roots. Suitable for advanced high school students and above.

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