📚 IGCSE Mathematics: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
In IGCSE Mathematics, quadratic equations are a central topic that connects algebra, graphing, and real-world problem solving. This article covers the key methods, graphical interpretation, and exam strategies you need to master quadratics with confidence.
在 IGCSE 数学中，二次方程是连接代数、图像和实际问题求解的核心主题。本文涵盖关键解法、图像解读以及考试策略，帮助你自信掌握二次方程。
1. What Is a Quadratic Equation? | 什么是二次方程？
e written in the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
二次方程是一个次数为 2 的多项式方程。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。
ar term, and c is the constant term. If a = 0, the equation becomes linear, not quadratic.
项 ax² 称为二次项，bx 称为一次项，c 称为常数项。如果 a = 0，方程就变成一次方程，不再是二次方程。
Quadratic equations can have zero, one, or two real solutions. These solutions are also called roots or x-intercepts of the corresponding quadratic function.
二次方程可以有零个、一个或两个实数解。这些解也称为对应二次函数的根或 x 轴截距。
2. Standard Form and Coefficients | 标准形式与系数
. This makes it easier to identify a, b, and c for factorisation or the quadratic formula.
在解二次方程之前，必须先将它整理为标准形式 ax² + bx + c = 0。这样可以更容易地识别 a、b、c，便于因式分解或使用公式法。
rearranged to -2x² + 3x - 5 = 0, or equivalently 2x² - 3x + 5 = 0 after multiplying by -1.
例如，3x - 2x² = 5 应整理为 -2x² + 3x - 5 = 0，或者乘以 -1 后等价为 2x² - 3x + 5 = 0。
+ 2)(x + 3) = 14 must be expanded to x² + 5x + 6 = 14, then rearranged to x² + 5x - 8 = 0.
有些方程需要先展开括号。例如 (x + 2)(x + 3) = 14 必须先展开为 x² + 5x + 6 = 14，再整理为 x² + 5x - 8 = 0。
Always check that the right-hand side is zero before applying any method. Missing this step is a common source of errors.
在使用任何方法之前，务必检查等式右边是否为零。忽略这一步是常见的错误来源。
3. Solving by Factorisation | 因式分解法
en as a product of two binomials. For example, x² + 5x + 6 = 0 becomes (x + 2)(x + 3) = 0.
当二次表达式可以写成两个二项式的乘积时，因式分解法通常是最快的方法。例如，x² + 5x + 6 = 0 可变为 (x + 2)(x + 3) = 0。
en at least one factor must be zero. Thus x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.
根据零乘积性质，如果乘积为零，那么至少有一个因式为零。因此 x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。
When the leading coefficient is not 1, such as 2x² + 7x + 3, you may need to split the middle term or use trial and error to find factors like (2x + 1)(x + 3).
当首项系数不为 1 时，例如 2x² + 7x + 3，你可能需要拆分中间项或通过尝试找到因式，如 (2x + 1)(x + 3)。
Recognising special patterns also helps. For example, the difference of squares x² - 9 can be written as (x + 3)(x - 3).
识别特殊模式也有帮助。例如，平方差 x² - 9 可以写成 (x + 3)(x - 3)。
4. Solving by Completing the Square | 配方法
Completing the square is a powerful method that works for any quadratic equation and is also used to derive the quadratic formula. The goal is to rewrite the expression as a perfect square plus a constant.
配方法是一种适用于任何二次方程的有效方法，也用于推导求根公式。其目标是将表达式改写为一个完全平方加上一个常数。
tant: x² + 6x = -5. Then add (6 ÷ 2)² = 9 to both sides: x² + 6x + 9 = 4, so (x + 3)² = 4.
对于 x² + 6x + 5 = 0，先将常数移项：x² + 6x = -5。然后两边加上 (6 ÷ 2)² = 9：x² + 6x + 9 = 4，因此 (x + 3)² = 4。
Finally, take square roots: x + 3 = ±√4, giving x + 3 = 2 or x + 3 = -2, so x = -1 or x = -5.
最后开平方：x + 3 = ±√4，
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