Zero Product Property - Algebra-1

1. Fundamental Concepts

The Zero Product Property is an important tool for solving quadratic equations. Its core content is: If the product of two or more factors equals 0, then at least one of the factors must be 0.
Expressed in mathematical language: If \(A \times B = 0\), then either \(A = 0\) or \(B = 0\) (or both are 0).
This property is the key basis for converting quadratic equations into linear equations for solution. By factoring and then using this property, we can quickly find the solutions to the equations.

Application Scenarios: It is mainly used to solve equations that can be factored into the product of factors, especially quadratic equations (such as \(ax^2 + bx + c = 0\) factored into the form of \((mx + n)(px + q) = 0\)).
Logical Essence: It is based on the multiplication rule that "0 multiplied by any number is 0". Conversely, if the product is 0, at least one of the multipliers must be 0.
Operational Premise: The equation needs to be rearranged into the form of "product of factors = 0", that is, one side of the equal sign is 0, and the other side is the factored expression.

Easy Level
Equation: \((3x - 6)(5x + 20) = 0\) (in the form of the product of two factors)

Solution steps:
1. Apply the Zero Product Property (ZPP) and set each factor equal to 0 separately:
  - \(3x - 6 = 0\)
  - \(5x + 20 = 0\)
2. Solve each of the above linear equations respectively:
  - From \(3x - 6 = 0\), we get \(3x = 6\), that is, \(x = 2\);
  - From \(5x + 20 = 0\), we get \(5x = -20\), that is, \(x = -4\).
3. Therefore, the solutions to the equation are \(x = 2\) or \(x = -4\).
Medium Level
- Equation: \(x^2 - 5x + 6 = 0\)
- Steps:
  1. Factorization: \((x - 2)(x - 3) = 0\)
  2. Apply ZPP: \(x - 2 = 0\) or \(x - 3 = 0\)
  3. Solutions: \(x = 2\) or \(x = 3\)
Hard Level Equation: \(3x^2 - 11x - 4 = 0\)
Steps:
1. Factorization: \((3x + 1)(x - 4) = 0\)
2. Apply ZPP: \(3x + 1 = 0\) or \(x - 4 = 0\)
3. Solutions: \(x =-\frac{1}{3}\) or \(x = 4\)

Rearrange the equation form: Ensure that one side of the equation's equal sign is 0, and the other side is a polynomial (such as \(ax^2 + bx + c = 0\)).
Factorization: Factor the polynomial into the product of two or more factors (such as \((mx + n)(px + q)=0\)). If direct factorization is not possible, try the completing the square method to convert it into a square form.
Apply ZPP: Set each factor equal to 0 separately to get multiple linear equations.
Solve the linear equations: Solve each linear equation separately to obtain all solutions of the original quadratic equation.
Verify the correctness of the solutions: Substitute the solutions back into the original equation to check if the equation holds (an optional step to ensure the calculation is correct).