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The equation provided is:

\[
\sqrt{x + 2} + \sqrt{7x + 2} = 6
\]

### Solution:

Step 1: Isolate one of the square roots.
\[
\sqrt{x + 2} = 6 - \sqrt{7x + 2}
\]

Step 2: Square both sides to eliminate the square root on the left.
\[
x + 2 = (6 - \sqrt{7x + 2})^2
\]
Expanding the right-hand side:
\[
x + 2 = 36 - 12\sqrt{7x + 2} + (7x + 2)
\]
Simplifying:
\[
x + 2 = 38 + 7x - 12\sqrt{7x + 2}
\]
Step 3: Isolate the remaining square root.
\[
x + 2 - 38 - 7x = -12\sqrt{7x + 2}
\]
\[
-6x - 36 = -12\sqrt{7x + 2}
\]
Divide by -12 on both sides:
\[
\frac{6x + 36}{12} = \sqrt{7x + 2}
\]
\[
\frac{x}{2} + 3 = \sqrt{7x + 2}
\]

Step 4: Square both sides again.
\[
\left(\frac{x}{2} + 3\right)^2 = 7x + 2
\]
Expanding the left-hand side:
\[
\frac{x^2}{4} + 3x + 9 = 7x + 2
\]
Step 5: Rearrange the equation.
\[
\frac{x^2}{4} + 3x + 9 - 7x - 2 = 0
\]
Simplify:
\[
\frac{x^2}{4} - 4x + 7 = 0
\]
Multiply through by 4 to clear the fraction:
\[
x^2 - 16x + 28 = 0
\]

Step 6: Solve the quadratic equation.
Using the quadratic formula:
\[
x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(28)}}{2(1)}
\]
\[
x = \frac{16 \pm \sqrt{256 - 112}}{2}
\]
\[
x = \frac{16 \pm \sqrt{144}}{2}
\]
\[
x = \frac{16 \pm 12}{2}
\]
So, the two possible solutions are:
\[
x = \frac{16 + 12}{2} = 14 \quad \text{or} \quad x = \frac{16 - 12}{2} = 2
\]

Step 7: Check for extraneous solutions.
- For \(x = 14\):
\[
\sqrt{14 + 2} + \sqrt{7(14) + 2} = \sqrt{16} + \sqrt{100} = 4 + 10 = 14 \quad (\text{False, so discard this solution})
\]
- For \(x = 2\):
\[
\sqrt{2 + 2} + \sqrt{7(2) + 2} = \sqrt{4} + \sqrt{16} = 2 + 4 = 6 \quad (\text{True})
\]

Thus, the solution is:
\[
x = 2
\]

Would you like further details on any step?

### Follow-up Questions:
1. What are extraneous solutions and why do they occur?
2. Can this method be generalized for other square root equations?
3. What would happen if the equation had a cubic root instead of a square root?
4. How do you solve quadratic equations with complex roots?
5. Why is squaring both sides of an equation sometimes problematic?

**Tip:** Always check for extraneous solutions when dealing with square roots, as squaring both sides can introduce false solutions.

Learn how to solve the radical equation sqrt(x + 2) + sqrt(7x + 2) = 6 with step-by-step instructions. This problem involves simplifying square roots and solving quadratic equations. Includes tips on checking for extraneous solutions. Suitable for high school students in Grades 10-12.

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