Certainly! I'll solve the questions step-by-step. Here’s the breakdown:

### Questions:
1. \( x^2 - 4x + 3 = 0 \)
2. \( 2y^2 + 3y - 5 = 0 \)
3. \( 3x^2 - 2x - 1 = 0 \)
4. \( x^2 - 4 = 0 \)
5. \( 4y^2 - 12y + 9 = 0 \)
6. \( 3x^2 + 7x + 2 = 0 \)
7. \( 4x^2 + 4x - 3 = 0 \)
8. \( y^2 - 6y + 9 = 0 \)
9. \( 2x^2 - 8x + 6 = 0 \)
10. \( 2y^2 - y - 6 = 0 \)

### Solutions:

1. **\( x^2 - 4x + 3 = 0 \)**
   - **Factorization:**
     \[
     x^2 - 4x + 3 = (x - 1)(x - 3) = 0
     \]
   - **Solutions:** \( x = 1 \), \( x = 3 \).

2. **\( 2y^2 + 3y - 5 = 0 \)**
   - **Quadratic Formula:**
     \[
     y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
     \]
     Substituting \(a = 2\), \(b = 3\), \(c = -5\):
     \[
     y = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4}
     \]
   - **Solutions:** \( y = 1 \), \( y = -\frac{5}{2} \).

3. **\( 3x^2 - 2x - 1 = 0 \)**
   - **Quadratic Formula:**
     \[
     x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-1)}}{2(3)} = \frac{2 \pm \sqrt{4 + 12}}{6} = \frac{2 \pm \sqrt{16}}{6}
     \]
   - **Solutions:** \( x = 1 \), \( x = -\frac{1}{3} \).

4. **\( x^2 - 4 = 0 \)**
   - **Factorization:**
     \[
     x^2 - 4 = (x - 2)(x + 2) = 0
     \]
   - **Solutions:** \( x = 2 \), \( x = -2 \).

5. **\( 4y^2 - 12y + 9 = 0 \)**
   - **Perfect Square Trinomial:**
     \[
     4y^2 - 12y + 9 = (2y - 3)^2 = 0
     \]
   - **Solution:** \( y = \frac{3}{2} \).

6. **\( 3x^2 + 7x + 2 = 0 \)**
   - **Quadratic Formula:**
     \[
     x = \frac{-7 \pm \sqrt{7^2 - 4(3)(2)}}{2(3)} = \frac{-7 \pm \sqrt{49 - 24}}{6} = \frac{-7 \pm \sqrt{25}}{6}
     \]
   - **Solutions:** \( x = -\frac{1}{3} \), \( x = -2 \).

7. **\( 4x^2 + 4x - 3 = 0 \)**
   - **Quadratic Formula:**
     \[
     x = \frac{-4 \pm \sqrt{4^2 + 4(3)(4)}}{2(4)} = \frac{-4 \pm \sqrt{16 + 48}}{8} = \frac{-4 \pm \sqrt{64}}{8}
     \]
   - **Solutions:** \( x = \frac{1}{2} \), \( x = -1 \).

8. **\( y^2 - 6y + 9 = 0 \)**
   - **Perfect Square Trinomial:**
     \[
     y^2 - 6y + 9 = (y - 3)^2 = 0
     \]
   - **Solution:** \( y = 3 \).

9. **\( 2x^2 - 8x + 6 = 0 \)**
   - **Divide by 2:**
     \[
     x^2 - 4x + 3 = 0
     \]
     \[
     (x - 1)(x - 3) = 0
     \]
   - **Solutions:** \( x = 1 \), \( x = 3 \).

10. **\( 2y^2 - y - 6 = 0 \)**
    - **Quadratic Formula:**
      \[
      y = \frac{-(-1) \pm \sqrt{(-1)^2 + 4(2)(6)}}{2(2)} = \frac{1 \pm \sqrt{1 + 48}}{4} = \frac{1 \pm \sqrt{49}}{4}
      \]
    - **Solutions:** \( y = 2 \), \( y = -\frac{3}{2} \).

Would you like more details on any of these steps or do you have further questions?

### Related Questions:
1. How can you determine if a quadratic equation can be factored easily?
2. What is the significance of the discriminant in a quadratic equation?
3. How do you solve quadratic equations using the completing the square method?
4. What are the key differences between solving quadratic equations by factoring versus using the quadratic formula?
5. Can all quadratic equations be solved by factoring? If not, why?

**Tip:** Always simplify the equation if possible before solving; for example, dividing by a common factor can make the problem easier to manage.

Explore how to solve various quadratic equations step-by-step. Includes detailed solutions for equations like x^2 - 4x + 3 = 0, 2y^2 + 3y - 5 = 0, and more. Learn key methods such as factoring, using the quadratic formula, and recognizing perfect square trinomials.

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