📚 Mastering Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 掌握二次方程：因式分解、配方法与求根公式
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to coordinate geometry and real-world modelling. This article explains the core methods for solving quadratic equations, the meaning of the discriminant, and how to interpret quadratic graphs. Work through each section carefully and test yourself with the embedded examples.
二次方程贯穿 IGCSE 数学课程，从代数运算到坐标几何和实际问题建模均有涉及。本文讲解解二次方程的核心方法、判别式的含义以及如何解读二次函数的图像。请仔细学习每一节，并使用文中的例题进行自测。
1. What is a Quadratic Equation? | 什么是二次方程？
d a ≠ 0. The highest power of the variable x is 2, which gives the equation its characteristic U-shaped graph called a parabola.
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。变量 x 的最高次数为 2，因此它的图像是一条特有的 U 形曲线，称为抛物线。
e linear, not quadratic. For example, x² - 4 = 0 is quadratic, while 2x - 4 = 0 is linear.
条件 a ≠ 0 很关键：如果 a 为零，x² 项就会消失，方程就变成一次方程而不是二次方程。例如，x² - 4 = 0 是二次方程，而 2x - 4 = 0 是一次方程。
2. Standard Form and Key Terminology | 标准形式与关键术语
0 after subtracting 2x from both sides. Rearranging correctly prevents sign errors later.
在求解之前，务必先把方程整理成标准形式 ax² + bx + c = 0。例如，3x² - 5 = 2x 两边减去 2x 后得到 3x² - 2x - 5 = 0。正确移项可以避免后续出现符号错误。
quadratic formula and discriminant. In 3x² - 2x - 5 = 0, we have a = 3, b = -2 and c = -5.
系数 a 是 x² 项的系数，系数 b 是 x 项的系数，c 是常数项。正确识别 a、b 和 c 对使用求根公式和判别式至关重要。在 3x² - 2x - 5 = 0 中，a = 3，b = -2，c = -5。
3. Solving by Factorising | 因式分解法
numbers that multiply to 6 and add to 5: 2 and 3. Hence x² + 5x + 6 = (x + 2)(x + 3) = 0.
当二次方程有整数根时，因式分解通常是最快的方法。解 x² + 5x + 6 = 0 时，找出两个数，它们相乘得 6、相加得 5：2 和 3。因此 x² + 5x + 6 = (x + 2)(x + 3) = 0。
es x = -3. Always write your final answers as x = -2 or x = -3, and check by substitution.
令每个括号等于零：x + 2 = 0 得 x = -2，x + 3 = 0 得 x = -3。最终答案务必写成 x = -2 或 x = -3，并通过代入进行检验。
6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. The roots are x = -½ and x = -3.
当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，可以拆中项：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。根为 x = -½ 和 x = -3。
4. Solving by Completing the Square | 配方法
first write x² + 6x = -1. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 8, so (x + 3)² = 8.
配方法将 ax² + bx + c 转化为 a(x + p)² + q 的形式，不仅能直接看出顶点，也能用于求解。对于 x² + 6x + 1 = 0，先写成 x² + 6x = -1。两边加上 (6/2)² = 9，得 x² + 6x + 9 = 8，所以 (x + 3)² = 8。
ake the square root of both sides, remembering both positive and negative roots: x + 3 = ±√8 = ±2√2. Therefore x = -3 ± 2√2. These are the exact solutions in surd form.
对两边开平方，切记要取正负两个根：x + 3 = ±√8 = ±2√2。因此 x = -3 ± 2√2。这些是根式形式的精确解。
x + 2)² - 8 = -5, giving 2(x + 2)² = 3. Divide by 2 and take square roots to find x = -2 ± √3/2.
如果 x² 的系数不是 1，先将其提出。对于 2x² + 8x + 5 = 0，写成 2(x² + 4x) = -5，然后在括号内完成配方：2[(x + 2)² - 4] = -5，得 2(x + 2)² - 8 = -5，因此 2(x + 2)² = 3。除以 2 并开平方，得 x = -2 ± √3/2。
5. The Quadratic Formula | 求根公式
is especially useful when factorising is difficult or when coefficients are not integers.
求根公式可以求解任何标准形式 ax² + bx + c = 0 的二次方程。当因式分解困难或系数不是整数时，它尤其有用。
x = (-b ± √b² - 4ac) / (2a)
The expression b² - 4ac inside the square root is called the discriminant, written as Δ. It determines the number and type of roots of the equation.
平方根内的表达式 b²
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