To solve the expression \(\sqrt{7 + \sqrt{7 + \sqrt{7 + \sqrt{7 + \cdots}}}}\), we need to recognize it as an infinite nested radical. Let's assign the entire expression to a variable \(x\). Thus, we have:

\(x = \sqrt{7 + \sqrt{7 + \sqrt{7 + \cdots}}}\)

This implies:

\(x = \sqrt{7 + x}\)

To remove the square root, we square both sides of the equation:

\(x^2 = 7 + x\)

Rearranging the terms gives us a quadratic equation:

\(x^2 - x - 7 = 0\)

We can solve this quadratic equation using the quadratic formula, where \(a = 1\), \(b = -1\), and \(c = -7\). The quadratic formula is:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Substitute the values into the formula:

\(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \times 1 \times (-7)}}{2 \times 1}\)

Calculating further gives:

\(x = \frac{1 \pm \sqrt{1 + 28}}{2}\)

Simplifying under the square root:

\(x = \frac{1 \pm \sqrt{29}}{2}\)

The solutions to this equation are:

• \(x = \frac{1 + \sqrt{29}}{2}\)
• \(x = \frac{1 - \sqrt{29}}{2}\)

Since \(x\) represents a positive real number due to the context of a square root, we choose the solution:

\(x = \frac{1 + \sqrt{29}}{2}\)

Therefore, the correct answer is \(\frac{1 \pm \sqrt{29}}{2}\).