📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graph sketching and real-world modelling. Mastering the three main solution methods - factorising, completing the square, and the quadratic formula - gives you a reliable toolkit for any exam question. This revision guide explains each method step by step, connects the algebra to the graph, and highlights the most common pitfalls to avoid.
二次方程贯穿 IGCSE 数学课程，从代数运算到图像绘制和实际建模都会涉及。掌握三种主要解法——因式分解、配方法和求根公式——能为你应对任何考试题提供可靠的工具。本复习指南逐步讲解每种方法，将代数与图像联系起来，并强调需要避免的常见错误。
1. What Is a Quadratic Equation? | 什么是二次方程？
are constants and a ≠ 0. The highest power of x is 2, which is why it is called quadratic.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数，且 a ≠ 0。x 的最高次数是 2，因此称为二次方程。
ax² + bx + c = 0, a ≠ 0
e x represents an unknown number, and the equation may have two, one, or no real solutions.
如果 a = 0，方程就变成一元一次方程，而不是二次方程。例如 3x² - 5x + 2 = 0 中 a = 3，b = -5，c = 2。变量 x 表示未知数，方程可能有两个、一个或没有实数解。
2. Standard Form and Key Terms | 标准形式与关键术语
y a, b and c, and to choose the best method. Move all terms to one side and simplify if necessary.
求解前，一定要先把方程整理成标准形式 ax² + bx + c = 0。这样更容易识别 a、b 和 c，也更容易选择最佳方法。将所有项移到同一边，必要时进行化简。
be zero, but b or c can be zero, giving special cases such as ax² + c = 0 or ax² + bx = 0.
ax² 是二次项，bx 是一次项，c 是常数项。系数 a 不能为零，但 b 或 c 可以为零，从而得到如 ax² + c = 0 或 ax² + bx = 0 的特殊形式。
0. Identifying these cases early can save time because they are usually simpler to solve.
例如 2x² - 8 = 0 没有一次项，因此 b = 0。方程 x² + 6x = 0 没有常数项，因此 c = 0。尽早识别这些特殊情况可以节省时间，因为它们通常更容易求解。
3. Solving by Factorising | 因式分解法
Factorising works when the quadratic expression can be written as a product of two linear brackets. You set each bracket equal to zero and solve for x. This method is often the fastest when the roots are integers or simple fractions.
当二次式可以写成两个一次括号的乘积时，可使用因式分解法。你让每个括号等于零，然后解出 x。当根为整数或简单分数时，这种方法通常最快。
x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5. These are 2 and 3, so:
对于 x² + 5x + 6 = 0，找出两个数，使它们相乘得 6、相加得 5。这两个数是 2 和 3，因此：
x² + 5x + 6 = (x + 2)(x + 3) = 0
0, giving x = -2 or x = -3. Always expand your brackets mentally to check the middle term.
然后 x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。始终要在脑中展开括号来检验一次项。
numbers multiplying to a × c = 6 and adding to 7, namely 6 and 1. Then factor by grouping.
当 a ≠ 1 时，例如 2x² + 7x + 3，可以拆分一次项：找到两个数
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