📚 Mastering Quadratic Equations for IGCSE | IGCSE 二次方程完全攻略
A quadratic equation is one of the most important topics in the IGCSE Mathematics syllabus. It appears in algebra, graphs, and problem-solving questions, and it forms the foundation for more advanced work in AS and A Level Mathematics.
二次方程是 IGCSE 数学大纲中最重要的主题之一。它广泛出现在代数、图像和应用题中，也是后续 AS 与 A Level 数学学习的重要基础。
1. Understanding Quadratic Equations | 认识二次方程
and a ≠ 0. The condition a ≠ 0 is essential because if a = 0, the equation becomes linear.
一元二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。a ≠ 0 这一条件至关重要，因为若 a = 0，方程就退化为一元一次方程。
ax² + bx + c = 0, a ≠ 0
erm and c is the constant term. For example, 2x² - 5x + 3 = 0 has a = 2, b = -5 and c = 3.
在这个形式中，ax² 是二次项，bx 是一次项，c 是常数项。例如，2x² - 5x + 3 = 0 中 a = 2，b = -5，c = 3。
Some equations may not look quadratic at first. You may need to expand brackets, collect like terms, or move all terms to one side to reveal the standard form.
有些方程一开始看起来并不像二次方程。你可能需要展开括号、合并同类项，或将所有项移到一边，才能得到标准形式。
2. Solving by Factorising | 因式分解法
Factorising is usually the fastest method when the quadratic expression can be written as a product of two linear factors. We look for two numbers that multiply to give ac and add to give b, then split the middle term or use direct inspection.
当二次式可以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。我们需要找到两个数，它们的乘积等于 ac，和等于 b，然后拆项或直接观察分解。
d add to -5, we have (x - 2)(x - 3) = 0. Either x - 2 = 0 or x - 3 = 0, so x = 2 or x = 3.
例如，解 x² - 5x + 6 = 0。因为 -2 与 -3 的乘积为 6，和为 -5，所以 (x - 2)(x - 3) = 0。于是 x - 2 = 0 或 x - 3 = 0，因此 x = 2 或 x = 3。
x² - 5x + 6 = (x - 2)(x - 3) = 0 ⇒ x = 2 or x = 3
dd to 7. Split the middle term: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
当 a 不等于 1 时，因式分解需要更加仔细。对于 2x² + 7x + 3，ac = 6，数字 6 与 1 相加得 7。拆分一次项：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。
Always check your factors by expanding them back. This prevents sign errors and shows that the product equals the original quadratic expression.
一定要通过重新展开来检验因式分解，这可以防止符号错误，并验证乘积确实等于原二次式。
3. Solving by Completing the Square | 配方法
method is especially useful when the quadratic does not factorise neatly and when you need to find the vertex of a parabola.
配方法将二次式改写为 a(x + p)² + q = 0 的形式。当二次式不能简便分解，或需要求抛物线顶点时，这种方法特别有用。
, square it to get 9, and add 9 to both sides: x² + 6x + 9 = 16. This gives (x + 3)² = 16.
对于 x² + 6x - 7 = 0，先把常数移到右边：x² + 6x = 7。取 6 的一半，平方得 9，两边同时加 9：x² + 6x + 9 = 16。于是 (x + 3)² = 16。
(x + 3)² = 16 ⇒ x + 3 = ±4 ⇒ x = 1 or x = -7
by a. For 2x² - 8x + 6 = 0, divide by 2 to get x² - 4x + 3 = 0, then complete the square.
如果 a 不等于 1，应先将每一项除以 a。例如 2x² - 8x + 6 = 0，先除以 2 得到 x² - 4x + 3 = 0，再进行配方。
Remember that taking the square root introduces both a positive and a negative solution. Missing the negative root is a very common IGCSE error.
请记住，开平方会同时产生正、负两个解。漏掉负根是 IGCSE 考试中非常常见的错误。
4. The Quadratic Formula | 求根公式
impossible. It is derived from completing the square on the general form ax² + bx + c = 0.
求根公式可以求解任何二次方程，即使因式分解困难或不可能实现。它由一般形式 ax² + bx + c = 0 通过配方法推导而来。
x = (-b ± √b² - 4ac) / 2a
y, especially when coefficients are negative. For 2x² - 3x - 5 = 0, a = 2, b = -3, c = -5.
使用公式时，首先要从标准形式中确定 a、b 和 c。代入时要特别小心，尤其是系数为负的情况。例如 2x² - 3x - 5 = 0 中，a = 2，b = -3，c = -5。
x = (3 ± √9 + 40) / 4 = (3 ± 7) / 4 ⇒ x = 2.5 or x = -1
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