Let's go through solving each equation requested.

Problem 2

The equation given is: $x^2 - 7x + \sqrt{x^2 - 7x + 4} = 2$

To solve this, follow these steps:

1. Isolate the square root term: $\sqrt{x^2 - 7x + 4} = 2 - (x^2 - 7x)$

2. Square both sides to eliminate the square root: $x^2 - 7x + 4 = (2 - (x^2 - 7x))^2$ (Continue simplifying and solving for $x$.)

Problem 6

The equation given is: $\sqrt{x^2 + 4x + 1} + \sqrt{2x^2 + 8x + 9} = 4$

To solve this, follow these steps:

1. Isolate one square root term, for example: $\sqrt{x^2 + 4x + 1} = 4 - \sqrt{2x^2 + 8x + 9}$

2. Square both sides to eliminate the square root on the left side: $x^2 + 4x + 1 = (4 - \sqrt{2x^2 + 8x + 9})^2$ (Continue simplifying and solving for $x$.)

Would you like detailed steps for each equation from this point?

Here are 5 related questions for practice:

1. How would you solve an equation that has multiple square roots?
2. What happens if a square root equation has no solution?
3. What are the steps for verifying the solutions of a square root equation?
4. How can you apply factoring techniques when dealing with equations involving square roots?
5. Why is it necessary to square both sides of an equation with a square root?

Tip: Always check for extraneous solutions when squaring both sides of an equation, as this can introduce solutions that don’t satisfy the original equation.