📚 Quadratic Equations: Solving, Graphing and Applications | IGCSE 二次方程：求解、图像与应用
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic factorisation to graph sketching and real-life modelling. This revision guide brings together the methods you need to solve quadratics confidently, interpret their graphs, and avoid common exam errors.
二次方程贯穿 IGCSE 数学大纲，从基础因式分解到图像草图和实际建模。本复习指南汇总了你需要掌握的方法，帮助你自信地求解二次方程、理解图像并避免常见考试错误。
1. What is a Quadratic Equation? | 什么是二次方程？
c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数是 2。
For example, x² – 5x + 6 = 0 is a quadratic equation. However, x³ – 2x = 0 is not quadratic because it contains an x³ term.
例如，x² – 5x + 6 = 0 是二次方程。但 x³ – 2x = 0 不是二次方程，因为它含有 x³ 项。
Quadratic equations can have two real solutions, one repeated solution, or no real solutions. The number of solutions depends on the value of the discriminant, which we will explore later.
二次方程可以有两个实数解、一个重根解，或者没有实数解。解的个数取决于判别式的值，我们稍后会讨论。
2. Standard Form and Key Terms | 标准形式与关键术语
orising or using the quadratic formula. This makes it easy to identify the coefficients a, b and c correctly.
在因式分解或使用二次公式前，一定要把方程整理成标准形式 ax² + bx + c = 0。这样可以方便正确地识别系数 a、b 和 c。
e number multiplying x², b is the number multiplying x, and c is the constant term. In 3x² – 7x + 2 = 0, we have a = 3, b = –7 and c = 2.
系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。在 3x² – 7x + 2 = 0 中，a = 3，b = –7，c = 2。
ax² + bx + c = 0, a ≠ 0
Remember that a cannot be zero. If a = 0, the equation becomes linear, not quadratic.
记住 a 不能为 0。如果 a = 0，方程就变成一次方程，而不是二次方程。
3. Solving by Factorisation | 因式分解法求解
duct of two linear factors. If (px + q)(rx + s) = 0, then either px + q = 0 or rx + s = 0.
当二次式可以写成两个一次因式的乘积时，因式分解法很有效。如果 (px + q)(rx + s) = 0，则 px + q = 0 或 rx + s = 0。
For x² – 5x + 6 = 0, we need two numbers that multiply to 6 and add to –5. These numbers are –2 and –3. So (x – 2)(x – 3) = 0, giving x = 2 or x = 3.
对于 x² – 5x + 6 = 0，我们需要两个数，它们相乘为 6、相加为 –5。这两个数是 –2 和 –3。所以 (x – 2)(x – 3) = 0，得到 x = 2 或 x = 3。
test factor pairs of a and c. The correct factorisation is (2x + 1)(x + 3) = 0, giving x = –1/2 or x = –3.
当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，要尝试 a 和 c 的因数组合。正确因式分解为 (2x + 1)(x + 3) = 0，得到 x = –1/2 或 x = –3。
Always expand your factors to check them. A quick mental expansion can prevent sign errors in the exam.
始终把因式展开来检验。快速心算展开可以防止考试中的符号错误。
4. Solving by Completing the Square | 配方法求解
This is especially useful for finding turning points and when factorisation is difficult.
配方法把 ax² + bx + c = 0 改写为 (x + p)² = q 的形式。这在求顶点以及因式分解困难时特别有用。
For x² + 6x + 2 = 0, first rearrange to x² + 6x = –2. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 7. This gives (x +
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