1. Introduction: Turning One Squared Equation Into Two Straight Ones
You already know how to solve an equation like 3x + 5 = 0. That is a linear equation. The highest power of x is 1. One unknown, one answer.
Now look at this: 2x² - 5x + 3 = 0.
See the x²? That little 2 on top changes everything. This is a quadratic equation — an equation where the highest power of x is 2. The word "quadratic" comes from "quad", meaning square.
You cannot solve it by just shifting terms across the equals sign. That trick worked for linear equations. It will not work here. You need a new idea.
Here is the idea. If you can rewrite the left side as two brackets multiplied together — like (2x - 3)(x - 1) = 0 — then you are almost done. Why? Because if two things multiply to give zero, at least one of them must itself be zero. So one bracket is zero, or the other is. Each one gives you a value of x.
This whole section is about that one move: taking a quadratic and breaking it into two factors. A "factor" is something that divides into another thing exactly — just like 3 is a factor of 12. The method is called the factorisation method, and the key step inside it is called splitting the middle term.
You can now say what a quadratic equation is and why a fresh method is needed to solve it.