The key difference between a linear equation and a quadratic equation lies in their degree, graphical representation, and the number of possible solutions (x-intercepts).
Linear Equation
- Definition: A linear equation is an algebraic equation where the highest power of the variable is 1. It can be generally represented as ax + b = 0, where 'a' and 'b' are constants and 'x' is the variable.
- Graph: As stated in the reference, the graph of a linear equation is a line.
- X-Intercepts: Linear equations can have at most one x-intercept (where the line crosses the x-axis), as indicated by the reference.
- Example: 2x + 3 = 0
Quadratic Equation
- Definition: A quadratic equation is an algebraic equation where the highest power of the variable is 2. It can be generally represented as ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'x' is the variable (and a ≠ 0).
- Graph: As stated in the reference, the graph of a quadratic equation is a parabola, which is a U-shaped curve.
- X-Intercepts: Quadratic equations can have at most two x-intercepts (where the parabola crosses the x-axis), as the reference explains.
- Example: x² + 3x + 2 = 0
Summary Table
| Feature | Linear Equation | Quadratic Equation |
|---|---|---|
| Highest Power | 1 | 2 |
| General Form | ax + b = 0 | ax² + bx + c = 0 |
| Graph | Line | Parabola |
| X-Intercepts | At most one | At most two |
In essence, the fundamental difference is the degree of the variable. This difference leads to distinct graphical representations and potential solution sets for each type of equation.
