https://file.aiquickdraw.com/mathbot/file/1724471483094meori6xk.jpg

The uploaded image contains mathematical expressions in Thai. I will analyze the content, extract the mathematical problems, and provide solutions.

### Problems Identified:

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

2. **\( 2y^2 + 3y - 5 = 0 \)**
   \[
   \text{Using the quadratic formula: } y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   \[
   y = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm \sqrt{49}}{4}
   \]
   \[
   y = \frac{-3 \pm 7}{4}
   \]
   \[
   \text{Solutions: } y = 1, -\frac{5}{2}
   \]

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

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

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

6. **\( 3x^2 + 7x + 2 = 0 \)**
   \[
   \text{Using the quadratic formula: } x = \frac{-7 \pm \sqrt{49 - 24}}{6}
   \]
   \[
   x = \frac{-7 \pm \sqrt{25}}{6} = \frac{-7 \pm 5}{6}
   \]
   \[
   \text{Solutions: } x = -\frac{1}{3}, -2
   \]

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

8. **\( y^2 - 6y + 9 = 0 \)**
   \[
   \text{Factorization: } (y - 3)^2 = 0
   \]
   \[
   \text{Solution: } y = 3
   \]

9. **\( 2x^2 - 8x + 6 = 0 \)**
   \[
   \text{Divide the entire equation by 2: } x^2 - 4x + 3 = 0
   \]
   \[
   \text{Factorization: } (x - 1)(x - 3) = 0
   \]
   \[
   \text{Solutions: } x = 1, 3
   \]

10. **\( 2y^2 - y - 6 = 0 \)**
    \[
    \text{Using the quadratic formula: } y = \frac{1 \pm \sqrt{1 + 48}}{4}
    \]
    \[
    y = \frac{1 \pm \sqrt{49}}{4} = \frac{1 \pm 7}{4}
    \]
    \[
    \text{Solutions: } y = 2, -\frac{3}{2}
    \]

### Do you want details or have any questions?

### Related Questions:
1. How is the quadratic formula derived?
2. What are the different methods to solve quadratic equations?
3. What are the conditions for a quadratic equation to have real and distinct roots?
4. How does factorization work for solving quadratic equations?
5. What are the applications of quadratic equations in real life?

**Tip:** When solving quadratic equations, always check the discriminant \(b^2 - 4ac\) to determine the nature of the roots before proceeding.

Learn how to solve multiple quadratic equations including x^2 - 4x + 3 and 2y^2 + 3y - 5 using factorization and the quadratic formula. This detailed solution covers key algebraic concepts and is suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation