Solve Quadratics - When the Independent Term is Zero - Algebra-1

1. Fundamental Concepts

Definition: The general form of a quadratic equation is (). When the independent term (constant term) , the equation simplifies to .
Core Feature: The equation contains no constant term, and all terms include the variable x. Thus, it can be solved by factoring out the common factor.

Simplified Equation Form: (, , ).
Solution Principle: By factoring out the common factor x, the equation is transformed into $x(ax + b) = 0$. Using the property that "if the product of two factors is 0, at least one of the factors must be 0," we get:
- or
- Solving these gives two roots: and .
Characteristic of Roots: One root is always , and the other root is determined by the coefficients of the linear term and the quadratic term.

Identify the Equation Type: Confirm that the constant term in the quadratic equation, i.e., the equation is in the form .
Factor Out the Common Factor: Extract the common factor containing x to transform the equation into the product form .
Solve for Each Factor: Set each factor equal to 0 separately and solve for the two roots (one of which is always 0).
Verify the Results (optional): Substitute the solutions back into the original equation to check if they satisfy the equality (to ensure correctness).