The image contains two sets of problems involving square root equations. Let me guide you through solving a few equations from each set and explain the method.

Set 4: Solve these equations.

Example: Solve $\sqrt{2x - 1} = x$

1. Square both sides to remove the square root: $2x - 1 = x^2$
2. Rearrange the equation to set it equal to zero: $x^2 - 2x + 1 = 0$
3. Factor or solve the quadratic equation: $(x - 1)^2 = 0 \quad \Rightarrow \quad x = 1$
4. Check the solution by substituting back into the original equation: $\sqrt{2(1) - 1} = 1 \quad \Rightarrow \quad \sqrt{1} = 1$ The solution $x = 1$ is valid.

Set 5: Solve these equations.

Example: Solve $\sqrt{2x + 7} = \sqrt{x + 3} + 1$

1. Isolate one square root term: $\sqrt{2x + 7} - \sqrt{x + 3} = 1$
2. Square both sides to remove the square roots: $(\sqrt{2x + 7} - \sqrt{x + 3})^2 = 1^2$ Expanding the left-hand side: $(2x + 7) - 2\sqrt{(2x + 7)(x + 3)} + (x + 3) = 1$ Simplify: $3x + 10 - 2\sqrt{(2x + 7)(x + 3)} = 1$
3. Isolate the square root term: $3x + 9 = 2\sqrt{(2x + 7)(x + 3)}$
4. Square both sides again: $(3x + 9)^2 = 4(2x + 7)(x + 3)$ Expand and solve this quadratic equation for $x$.

Would you like detailed solutions for other specific equations, or explanations on how to approach the rest? Here are five related questions to deepen your understanding:

1. How do you handle extraneous solutions when solving square root equations?
2. What are the general steps for solving radical equations?
3. How do you check if a solution is valid after solving a square root equation?
4. Can square root equations have no real solutions? Why or why not?
5. What is the significance of squaring both sides when solving radical equations?

Tip: Always substitute your solutions back into the original equation to verify, as squaring both sides can introduce extraneous solutions.