📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程
Quadratic equations lie at the heart of IGCSE algebraic problem solving. Whether you are factorising, completing the square or using the quadratic formula, these equations test your ability to manipulate algebraic expressions and interpret roots in context. This guide covers the key definitions, methods and exam techniques you need for a confident performance.
二次方程是 IGCSE 代数问题求解的核心。无论是因式分解、配方法还是使用二次公式，这些方程都考查你操作代数表达式和在实际情境中解释根的能力。本指南涵盖你需要的重要定义、方法和应试技巧，帮助你自信应对考试。
1. What Is a Quadratic Equation? | 什么是二次方程？
n in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The word 'quadratic' comes from the Latin word 'quadratus', meaning square, because the highest power of the unknown x is 2.
二次方程是能写成一般形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数且 a ≠ 0。'quadratic' 一词源自拉丁语 'quadratus'，意为平方，因为未知数 x 的最高次数为 2。
ax² + bx + c = 0, a ≠ 0
If a = 0, the x² term disappears and the equation becomes linear rather than quadratic.
如果 a = 0，x² 项消失，方程就变成一次方程而不是二次方程。
2. Standard Form and Coefficients | 标准形式与系数
the coefficients a, b and c correctly is essential, especially when using the quadratic formula.
在解二次方程之前，你应始终把它整理成标准形式：ax² + bx + c = 0。正确识别系数 a、b 和 c 非常重要，尤其是在使用二次公式时。
we have a = 5, b = -3 and c = 8. Notice that the minus sign belongs to the coefficient b.
例如，在 5x² - 3x + 8 = 0 中，a = 5，b = -3，c = 8。请注意负号属于系数 b。
uation 2x² = 5x - 3 becomes 2x² - 5x + 3 = 0 in standard form, so a = 2, b = -5 and c = 3.
有时你必须先移项整理。方程 2x² = 5x - 3 变成标准形式 2x² - 5x + 3 = 0，因此 a = 2，b = -5，c = 3。
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets. The key principle is the zero product property: if two factors multiply to give zero, at least one of them must be zero.
当二次式能写成两个一次因式的乘积时，因式分解法通常是最快的方法。其核心原理是零积性质：如果两个因式的乘积为零，那么其中至少有一个因式必须为零。
or two numbers that multiply to give 12 and add to give -7: they are -3 and -4. Therefore:
解方程 x² - 7x + 12 = 0。我们寻找两个数，使它们的积为 12、和为 -7：这两个数是 -3 和 -4。因此：
x² - 7x + 12 = (x - 3)(x - 4) = 0
Setting each factor to zero gives x - 3 = 0 or x - 4 = 0, so x = 3 or x = 4.
令每个因式为零，得到 x - 3 = 0 或 x - 4 = 0，所以 x = 3 或 x = 4。
- Write the equation in standard form first.
- 先写成标准形式。
- Factorise fully into two brackets.
- 把二次式完全分解成两个括号的乘积。
- Set each bracket equal to zero and solve.
- 令每个括号等于零并求解。
4. Difference of Two Squares | 平方差公式
They may be a difference of two squares, which factorises using a² - b² = (a - b)(a + b).
有些二次方程没有中间的 x 项。它们可能是平方差，可以用公式
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