Quadratic Equations: Definition & Incomplete Forms
📐 What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in one variable. It is one of the most important types of equations in all of mathematics.
Standard (General) Form
$ax^2 + bx + c = 0$
where $a$, $b$, and $c$ are real numbers and $a \neq 0$
$a$ — Leading Coefficient
The coefficient of $x^2$. It cannot be zero (otherwise the equation wouldn't be quadratic). It determines the "width" and direction of the parabola.
$b$ — Linear Coefficient
The coefficient of $x$. It can be any real number, including zero. It affects the position of the parabola's axis of symmetry.
$c$ — Constant Term
The free term (no variable). It can be any real number, including zero. It represents the y-intercept of the parabola.
⚠️ Key Point: The word "quadratic" comes from the Latin word quadratus, meaning "square." The highest power of the variable must be exactly 2.
🔍 Identifying Quadratic Equations
| Equation | Quadratic? | Reason |
|---|---|---|
| $3x^2 + 5x - 2 = 0$ | ✅ Yes | $a=3$, $b=5$, $c=-2$; highest power is 2 |
| $x^2 - 9 = 0$ | ✅ Yes | $a=1$, $b=0$, $c=-9$; highest power is 2 |
| $5x + 3 = 0$ | ❌ No | Highest power is 1 (linear equation) |
| $0 \cdot x^2 + 4x - 1 = 0$ | ❌ No | $a = 0$, so it's actually linear: $4x - 1 = 0$ |
| $-x^2 + 7x = 0$ | ✅ Yes | $a=-1$, $b=7$, $c=0$; highest power is 2 |