https://file.aiquickdraw.com/mathbot/file/17269210468373hxwq2a4.jpg

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.

Solve the given set of square root equations from the uploaded image.

Explore step-by-step solutions for solving square root and radical equations. This guide includes key concepts from algebra and quadratic equations, with solutions suitable for students in Grades 9-11.

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