Concept 4.1
What Makes an Equation Quadratic
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
A quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
The intuition
The formula is not the real idea. What makes an equation quadratic is that the squared term survives after everything has been simplified. That term creates curvature: the graph bends.
This is why a ≠ 0 matters. If a becomes zero, the squared term vanishes and only the linear equation bx + c = 0 is left.
The attention
Look for the squared term that survives after simplifying.
x² + x = x² + 4 simplifies to x = 4
Look hereNot here
- Look here
- What is left after simplifying. Whether an equation is quadratic is decided after simplifying. Here the x² terms cancel and only x = 4 is left: linear.
- Not here
- The x² you can see before simplifying. An x² on the page proves nothing. It has to survive the simplification, which is exactly what a ≠ 0 guarantees.
- What it teaches
- Judge an expression by what it is after simplifying, not by how it first looks.