Find the value of √x + √x + √x +...........

The expression you're asking about is:

\[
\sqrt{x} + \sqrt{x} + \sqrt{x} + \cdots
\]

This suggests an infinite sum of square roots of \(x\), but it could also represent a continuous operation involving radicals. To clarify the interpretation:

1. **Sum of square roots**: If you're asking about adding \( \sqrt{x} \) repeatedly an infinite number of times, the sum would diverge unless \( x = 0 \).

2. **Nested square roots**: If you meant something like a **nested radical**, where:

\[
\sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}
\]

Then, we need to solve this as an equation. Let this expression be equal to \( y \):

\[
y = \sqrt{x + y}
\]

Now, square both sides to remove the square root:

\[
y^2 = x + y
\]

Rearrange the equation:

\[
y^2 - y - x = 0
\]

This is a quadratic equation in \( y \), so we can solve it using the quadratic formula:

\[
y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-x)}}{2(1)}
\]
\[
y = \frac{1 \pm \sqrt{1 + 4x}}{2}
\]

Since \( y \) must be positive, we take the positive root:

\[
y = \frac{1 + \sqrt{1 + 4x}}{2}
\]

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

\[
y = \frac{1 + \sqrt{1 + 4x}}{2}
\]

Would you like more details on this process?

---

Here are 5 related questions for further exploration:
1. What happens if \(x = 0\) in this nested radical equation?
2. Can this process be applied to different forms of nested radicals?
3. How does this solution compare to simpler radical equations like \( \sqrt{x + \sqrt{x}} \)?
4. What is the geometric interpretation of nested radicals?
5. How would the equation change if we added a constant inside the radical?

**Tip**: When solving equations with nested square roots, isolating the radical and squaring both sides is often the best strategy.

Explore how to solve an infinite nested radical expression of the form √x + √x + √x + ..., including step-by-step use of the quadratic formula. This detailed explanation covers algebraic concepts such as quadratic equations and is suitable for students in Grades 9-12.

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