Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, where the highest power of the variable is 2. It can be written in the form of ax^2 + bx + c = 0, where a, b, and c are constants, and 'x' represents the variable.

Factorization
Factorization is a method used to break down a quadratic equation into its factors. By finding the factors, we can determine the values of 'x' that satisfy the equation. The process involves expressing the quadratic equation as a product of two binomials.

Solving a Quadratic Equation by Factorization
To solve a quadratic equation by factorization, follow these steps:

1. Write the equation in the standard form: ax^2 + bx + c = 0.
2. Factorize the quadratic expression on the left side of the equation.
3. Set each factor equal to zero and solve for 'x'.
4. The values of 'x' obtained from step 3 are the solutions to the quadratic equation.

Example:
Let's solve the given quadratic equation by factorization: 2x/(x-3) + 1/2x + 3 - 3x + 9/(x-3)(2x + 3) = 0.

1. Combine like terms to simplify the equation:
2x/(x-3) + 1/2x - 3x + 3 + 9/(x-3)(2x + 3) = 0

2. Find a common denominator and simplify:
(4x^2 + (x - 3)(2x + 3) - 6x^2(x - 3) + 3(2x + 3) + 9) / ((x - 3)(2x + 3)) = 0

3. Expand and rearrange the equation:
(4x^2 + 2x^2 - 3x + 6x - 6x^3 + 18x^2 + 6x + 9) / ((x - 3)(2x + 3)) = 0

(-6x^3 + 24x^2 + 5x + 9) / ((x - 3)(2x + 3)) = 0

4. Factorize the numerator:
(-3x + 1)(2x + 3)(x - 3) / ((x - 3)(2x + 3)) = 0

5. Cancel out the common factors:
(-3x + 1) = 0

6. Solve for 'x':
-3x + 1 = 0
-3x = -1
x = 1/3

Conclusion
In this example, we solved a quadratic equation by factorization. By simplifying and rearranging the equation, we obtained a factorized form. Canceling out the common factors allowed us to solve for 'x' and find the solution to the quadratic equation.