📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式解法
d a ≠ 0. This article explains three algebraic methods for solving quadratics: factorising, completing the square and using the quadratic formula. It also covers the discriminant, graphical meaning and common exam errors.
二次方程是 IGCSE 数学大纲的核心内容。它们出现在代数、几何、测量和数据分析中。二次方程的一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。本文讲解解二次方程的三种代数方法：因式分解法、配方法和求根公式法，同时讨论判别式、图像意义以及常见考试错误。
1. Recognising Quadratic Equations | 认识二次方程
, and c is the constant term. If a = 0, the equation becomes linear rather than quadratic.
一元二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a ≠ 0。ax² 项称为二次项，bx 项称为一次项，c 称为常数项。如果 a = 0，方程就变成了一次方程，而不是二次方程。
also quadratic because it can be rearranged to x² − 9 = 0, where a = 1, b = 0 and c = −9.
例如，4x² − 3x + 7 = 0 是二次方程，其中 a = 4，b = −3，c = 7。方程 x² = 9 也是二次方程，因为它可以变形为 x² − 9 = 0，此时 a = 1，b = 0，c = −9。
mediately. For example, (x + 2)(x − 5) = 3 must be expanded and rearranged before solving.
并非所有含有 x² 的方程一开始都是标准形式。例如 (x + 2)(x − 5) = 3 需要先展开并整理后才能求解。
2. Solving by Factorising | 因式分解法解二次方程
When a quadratic expression can be written as the product of two linear factors, factorising is usually the fastest method. For the monic form x² + bx + c, find two numbers whose product is c and whose sum is b. These numbers become the constants in the factors.
当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。对于首项系数为 1 的形式 x² + bx + c，寻找两个数，使它们相乘得 c，相加得 b。这两个数就是因式中的常数。
with product 12 and sum 8. The numbers 2 and 6 work, so x² + 8x + 12 = (x + 2)(x + 6) = 0.
例：解 x² + 8x + 12 = 0。我们需要两个数，乘积为 12，和为 8。2 和 6 符合条件，因此 x² + 8x + 12 = (x + 2)(x + 6) = 0。
When a ≠ 1, use the ac method or splitting the middle term. Find two numbers whose product is ac and whose sum is b, then split the mx term and factor by grouping.
当 a ≠ 1 时，使用 ac 法或拆中项法。先找两个数，使它们相乘得 ac，相加得 b，然后拆分中项并分组分解。
ork. Therefore 3x² + 10x + 8 = 3x² + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2).
例：因式分解 3x² + 10x + 8。这里 ac = 3 × 8 = 24，需要和为 10。6 和 4 符合条件。因此 3x² + 10x + 8 = 3x² + 6x + 4x + 8 = 3x(x + 2) + 4(x + 2) = (3x + 4)(x + 2)。
3. The Zero Product Property | 零乘积性质
The zero product property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This is why a quadratic equation must be set equal to zero before factorising.
零乘积性质指出：如果两个或多个因式的乘积为零，那么至少有一个因式必须为零。这就是为什么在因式分解前必须把二次方程写成等于零的形式。
This gives the solutions x = 4 or x = −7. These are also called the roots of the equation.
若 (x − 4)(x + 7) = 0，则 x − 4 = 0 或 x + 7 = 0。由此得解 x = 4 或 x = −7，它们也称为方程的根。
g by x would remove the valid solution x = 0. Instead, set each factor to zero separately.
切勿把方程两边同除以含未知数的因式。例如在 x(x − 5) = 0 中，除以 x 会丢掉有效解 x = 0。正确做法是让每个因式分别等于零。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This is useful for solving equations, finding the vertex of a parabola, and deriving the quadratic formula.
配方法将二次式改写
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