Let's solve the equation \( 3x^2 = 5 - 2x \) and find the even numbers among the solutions.

1. **Rewrite the equation in standard form**:  
   \( 3x^2 + 2x - 5 = 0 \)

2. **Solve the quadratic equation** using the quadratic formula:  
   \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 2 \), and \( c = -5 \).

3. **Calculate the discriminant**:  
   \( b^2 - 4ac = 2^2 - 4(3)(-5) = 4 + 60 = 64 \)

4. **Find the roots**:  
   \( x = \frac{-2 \pm \sqrt{64}}{2(3)} \)  
   \( x = \frac{-2 \pm 8}{6} \)

5. **Calculate the two solutions**:  
   \( x_1 = \frac{-2 + 8}{6} = \frac{6}{6} = 1 \)  
   \( x_2 = \frac{-2 - 8}{6} = \frac{-10}{6} = -\frac{5}{3} \)

The solutions are \( x = 1 \) and \( x = -\frac{5}{3} \). Among these, the only integer value is \( x = 1 \). Since 1 is odd, there are no even integer solutions to the given equation.

Would you like a more detailed explanation of the steps or have any questions?

Here are some related questions for further exploration:

1. How do you determine if a quadratic equation has real or complex solutions?
2. What are the properties of quadratic equations with a discriminant of zero?
3. Can a quadratic equation have exactly one real root?
4. What is the sum of the roots of the quadratic equation \( 3x^2 + 2x - 5 = 0 \)?
5. How do you factorize a quadratic expression if the roots are not integers?
6. What are some practical applications of solving quadratic equations?
7. How does the graph of a quadratic function relate to its equation's solutions?
8. What is the axis of symmetry for the parabola represented by \( 3x^2 + 2x - 5 = 0 \)?

**Tip:** When using the quadratic formula, always check the discriminant first to determine the nature of the roots (real or complex).

Learn how to solve the quadratic equation 3x² = 5 - 2x to find integer solutions. This problem involves using the quadratic formula and understanding the discriminant to determine the nature of the roots. Suitable for students in Grades 9-12.

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