📚 Solving Quadratic Equations by Factorising, Completing the Square and the Quadratic Formula | 因式分解法、配方法和求根公式解二次方程
A quadratic equation appears throughout the IGCSE syllabus, from algebra and graphs to measurement problems. This revision guide explains the three main algebraic methods for finding exact solutions: factorising, completing the square, and using the quadratic formula.
二次方程贯穿 IGCSE 教学大纲，从代数、图像到实际测量问题都会出现。本复习指南讲解三种主要代数解法：因式分解法、配方法和求根公式法。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
being 2. The standard form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程的特点是变量的最高次数为 2。其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。
ax² + bx + c = 0, a ≠ 0
0, the equation becomes linear, not quadratic. The coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term.
条件 a ≠ 0 非常重要；如果 a = 0，该方程就变成一元一次方程而不是二次方程。系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。
lready in standard form. However, x² = 7x + 8 must first be rearranged to x² - 7x - 8 = 0.
像 2x² + 5x - 3 = 0 和 x² - 4 = 0 这样的方程已经是标准形式。但 x² = 7x + 8 必须首先整理为 x² - 7x - 8 = 0。
Always move all terms to one side so that the other side equals zero before attempting any solution method.
在尝试任何解法之前，务必把所有项移到一边，使另一边等于零。
2. Solving by Factorising | 因式分解法
ion can be written as a product of two linear factors. This is usually easiest when a = 1.
当二次式可以写成两个一次因式的乘积时，因式分解法是最快的方法。这通常在 a = 1 时最容易。
-6 and add to -1. Those numbers are -3 and 2, so the equation becomes (x - 3)(x + 2) = 0.
例如，要解 x² - x - 6 = 0，我们需要两个数相乘得 -6、相加得 -1。这两个数是 -3 和 2，因此方程变为 (x - 3)(x + 2) = 0。
+ 7x + 3 = 0, we find two numbers that multiply to 2 × 3 = 6 and add to 7, namely 6 and 1.
当 a ≠ 1 时，因式分解过程也需要用到乘积 ac 与和 b。对于 2x² + 7x + 3 = 0，我们要找两个数相乘得 2 × 3 = 6、相加得 7，这两个数是 6 和 1。
x, then factor by grouping: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0.
将中间项改写为 6x + x，然后分组分解：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。
Factorising is not always possible using integers, which is why other methods are needed.
并非所有二次式都能用整数进行因式分解，因此还需要其他方法。
3. The Zero Product Property | 零乘积性质
y: if mn = 0, then m = 0 or n = 0. This means at least one of the factors must equal zero.
因式分解法成立的依据是零乘积性质：如果 mn = 0，那么 m = 0 或 n = 0。这意味着至少有一个因式等于零。
+ 2) = 0 we set each factor equal to zero: x - 3 = 0 or x + 2 = 0, giving x = 3 or x = -2.
由 (x - 3)(x + 2) = 0，我们令每个因式等于零：x - 3 = 0 或 x + 2 = 0，得到 x = 3 或 x = -2。
example, starting with x² = 4x, dividing by x gives x = 4, but the solution x = 0 is lost.
一个非常常见的错误是两边同时除以含未知数的因式。例如，从 x² = 4x 出发，除以 x 得到 x = 4，但这样丢失了 x = 0 这个解。
oach is to rearrange to x² - 4x = 0, factorise as x(x - 4) = 0, and obtain x = 0 or x = 4.
正确的做法是整理为 x² - 4x = 0，分解为 x(x - 4) = 0，得到 x = 0 或 x = 4。
Never divide by an expression containing the variable unless you are certain it cannot be zero.
除非你能确定含未知数的表达式不会为零，否则不要用它去除方程两边。
4. Completing the Square: Concept | 配方法的概念
Completing the square rewrites a quadratic expression in the form (x + p)² + q, which is useful when factorising is difficult or when we need the vertex of a parabola.
配方法将二次式改写为 (x + p)² + q 的形式，这在因式分解困难或需要求抛物线顶点时非常有用。
+ 2px + p² = (x + p)². To complete x² + bx, we add (b/2)² because b plays the role of 2p.
这一思想来源于恒等式 x² + 2px + p² = (x + p)
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