📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式
ng. A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a ≠ 0. This article explains four essential methods: factorising, completing the square, using the quadratic formula, and interpreting the discriminant.
二次方程贯穿整个 IGCSE 数学大纲，从代数、问题求解到图像和现实建模都会出现。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 ≠ 0。本文讲解四种核心方法：因式分解、配方法、使用求根公式以及判别式的解读。
1. Recognising a Quadratic Equation | 识别二次方程
he equation would become linear. For example, 2x² + 5x - 3 = 0 is quadratic, while 3x + 1 = 0 is linear.
二次方程中 x 的最高次数是 2。标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不能为零。如果 a 为零，方程就变成了一次方程。例如，2x² + 5x - 3 = 0 是二次方程，而 3x + 1 = 0 是一次方程。
ms first. For instance, (x + 3)(x - 2) = 6 must be expanded and rearranged before solving.
有时二次方程不会以标准形式给出；你必须先展开括号或合并同类项。例如，(x + 3)(x - 2) = 6 必须先展开并整理后才能求解。
Key features to check: one variable, highest index 2, and a non-zero coefficient of x². If any of these fail, it is not a quadratic equation.
需要检查的关键特征：一个变量、最高指数为 2、x² 的系数非零。如果任一条件不满足，它就不是二次方程。
2. Standard Form and Rearranging | 标准形式与整理
Before choosing a method, always rewrite the equation so that one side is zero. This means expanding products, removing fractions, and collecting like terms. The coefficient a is the number in front of x², b is the number in front of x, and c is the constant term.
在选择方法之前，总是先重写方程，使一边为零。这意味着展开乘积、去掉分母并合并同类项。系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。
and c = -7. Missing terms still have a zero coefficient: x² - 9 = 0 has b = 0 and c = -9.
例如，3x² - 7 = 2x 变为 3x² - 2x - 7 = 0，所以 a = 3，b = -2，c = -7。缺失的项系数为零：x² - 9 = 0 中 b = 0，c = -9。
| Equation | 方程 | a | b | c |
|---|---|---|---|
| 2x² + 5x - 3 = 0 | 2 | 5 | -3 |
| x² - 9 = 0 | 1 | 0 | -9 |
| 3x² - 2x - 7 = 0 | 3 | -2 | -7 |
ominator. For example, x²/2 + x/3 - 1 = 0 becomes 3x² + 2x - 6 = 0 after multiplying by 6.
如果出现分数，将每一项乘以公分母。例如，x²/2 + x/3 - 1 = 0 乘以 6 后变为 3x² + 2x - 6 = 0。
3. Solving by Factorising | 因式分解法求解
a product of two brackets, then use the zero product rule: if AB = 0, then A = 0 or B = 0.
当二次方程能用整数因式分解时，因式分解通常是最快的方法。要解 ax² + bx + c = 0，把它写成两个括号的乘积，然后使用零积法则：如果 AB = 0，则 A = 0 或 B = 0。
+12 and add to +7. They are +3 and +4, so (x + 3)(x + 4) = 0. Therefore x = -3 or x = -4.
对于 x² + 7x + 12 = 0，找两个数，乘积为 +12，和为 +7。它们是 +3 和 +4，所以 (x + 3)(x + 4) = 0。因此 x = -3 或 x = -4。
x - 3 = 0, then 2x(x + 3) - 1(x + 3) = 0, so (2x - 1)(x + 3) = 0, giving x = ½ or x = -3.
对于 2x² + 5x - 3 = 0，拆分中间项：2x² + 6x - x - 3 = 0，然后 2x(x + 3) - 1(x + 3) = 0，所以 (2x - 1)(x + 3) = 0，得到 x = ½ 或 x = -3。
Always check by expanding the brackets. If the product does not match the original quadratic, the factorisation is wrong.
始终展开括号检查。如果乘积与原二次式不符，说明因式分解错误。
4. Completing the Square: The Method | 配方法：步骤
s x² + bx + c in the form (x + p)² + q. The key identity is x² + bx = (x + b/2)² - (b/2)².
配方法把 x² + bx + c 重写为 (x + p)² + q 的形式。关键恒等式是 x² + bx = (x + b/2)² - (b/2)²。
x² + bx + c = (x + b/2)² + c − (b/2)²
Steps: (1) factor out a if a ≠ 1; (2) take half of the coefficient of x; (3) square the bracket; (4) subtract the square of that half coefficient; (5) add the original constant.
步骤：(1) 如果
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图