Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear both as pure algebra questions and as tools for solving real-world problems, from projectile motion to area calculations. Mastering this topic requires understanding multiple methods — factorisation, completing the square, the quadratic formula, and graphical interpretation. This article will guide you through each method step by step, clarifying when to use which approach and how to avoid the most common pitfalls.

一元二次方程是IGCSE数学中最核心的考点之一。无论是纯代数题，还是从抛体运动到面积计算等实际应用，二次方程无处不在。掌握这一考点，需要理解多种解法——因式分解、配方法、求根公式和图像分析。本文将从零开始，逐步讲解每种方法、适用场景以及最常见的易错点，帮助你稳拿分数。

1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The word ‘quadratic’ comes from the Latin ‘quadratus’, meaning square, because the highest power of the variable is 2. A quadratic expression has degree 2, so it can have at most two distinct solutions.

一元二次方程是能写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。quadratic 一词源自拉丁语 quadratus，意为“平方”，因为变量的最高次数是 2。二次表达式的次数为 2，因此它至多有两个不同的解。

A solution of a quadratic equation is called a root. Every quadratic equation has exactly two roots if complex roots are counted; these roots may be distinct, equal, or non-real. The roots can be found by factorisation, completing the square, using the quadratic formula, or reading them from a graph.

二次方程的解称为根。若把复数根也算在内，每个一元二次方程恰好有两个根；这两个根可能不相等、相等，或者不是实数。求根可以通过因式分解、配方、使用求根公式，或者从图像中读出。

x² − 5x + 6 = 0

Here a = 1, b = −5 and c = 6. The roots are x = 2 and x = 3 because (x − 2)(x − 3) = 0. Substituting either value into the original equation gives zero, which is the check every candidate should perform.

这里 a = 1，b = −5，c = 6。两根为 x = 2 和 x = 3，因为 (x − 2)(x − 3) = 0。把任何一个值代回原方程都等于零——这是每位考生都应当做的验算。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic can be written as a product of two linear factors. It relies on the zero product property: if p × q = 0, then p = 0 or q = 0. Once a quadratic is factored as (x − m)(x − n) = 0, the roots are immediately m and n.

当二次式能写成两个一次因式的乘积时，因式分解是最快的方法。它依赖于零乘积性质：若 p × q = 0，则 p = 0 或 q = 0。一旦二次式分解为 (x − m)(x − n) = 0，根立即就是 m 和 n。

Step 1: Rearrange the equation so that one side is zero. 步骤一：整理方程，使一边为零。
Step 2: Factorise the quadratic expression fully. 步骤二：将二次表达式完全因式分解。
Step 3: Set each factor equal to zero and solve the linear equations. 步骤三：令每个因式等于零，并解这两个一次方程。

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3

When the coefficient a is not 1, look for factors that multiply to a × c and add to b. For example, in 2x² + 5x − 3 = 0, the two numbers are 6 and −1 because 6 × (−1) = −6 = a × c and 6 + (−1) = 5 = b. Then split the middle term: 2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3) = 0, so x = ½ or x = −3.

当 a 不为 1 时，寻找两个数，使其乘积等于 a × c、之和等于 b。例如，在 2x² + 5x − 3 = 0 中，这两个数是 6 和 −1，因为 6 × (−1) = −6 = a × c，且 6 + (−1) = 5 = b。然后拆中项：2x² + 6x − x − 3 = 2x(x + 3) − 1(x + 3) = (2x − 1)(x + 3) = 0，所以 x = ½ 或 x = −3。

3. Difference of Two Squares | 平方差公式

A special factorisation pattern that appears frequently in IGCSE exams is the difference of two squares: a² − b² = (a + b)(a − b). Any quadratic of the form x² − k² can be solved instantly with this identity, and it also works when the first term has a coefficient.

IGCSE 考试中经常出现一种特殊的因式分解模式——平方差公式：a² − b² = (a + b)(a − b)。任何形如 x² − k² 的二次方程都可以用这个恒等式立即求解，当首项带有系数时同样适用。

x² − 25 = 0 → (x + 5)(x − 5) = 0 → x = ±5

For a harder case, consider 4x² − 9 = 0. Rewrite it as (2x)² − 3² = 0. Then (2x + 3)(2x − 3) = 0, giving x = −3/2 or x = 3/2. A closely related pattern is the perfect square: x² ± 2kx + k² = (x ± k)². Recognising these patterns saves valuable exam time.

再看一个稍难的例子：4x² − 9 = 0。把它改写为 (2x)² − 3² = 0，则 (2x + 3)(2x − 3) = 0，得到 x = −3/2 或 x = 3/2。与之密切相关的是完全平方公式：x² ± 2kx + k² = (x ± k)²。在考试中识别这些模式可以节省宝贵时间。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + h)² + k. This method is essential because it reveals the turning point of the graph and is the foundation of the quadratic formula. When a question explicitly says ‘by completing the square’, you must use this method to earn full marks.

配方法将二次式改写为 a(x + h)² + k 的形式。这个方法至关重要，因为它能直接揭示图像的顶点，也是求根公式的基础。当题目明确要求“用配方法”时，必须使用此法才能拿到满分。

For an expression x² + bx, take half of b, square it, then add and subtract that value. For example, to solve x² + 6x + 2 = 0, first complete the square on x² + 6x. Half of 6 is 3, and 3² = 9, so x² + 6x = (x + 3)² − 9.

对于形如 x² + bx 的式子，取 b 的一半，将其平方，然后加上再减去这个值。例如，解 x² + 6x + 2 = 0，先对 x² + 6x 配方。6 的一半是 3，3² = 9，所以 x² + 6x = (x + 3)² − 9。

x² + 6x + 2 = 0 → (x + 3)² − 9 + 2 = 0 → (x + 3)² = 7

Taking square roots of both sides gives x + 3 = ±√7, so the two roots are x = −3 + √7 and x = −3 − √7. Notice the roots are irrational; completing the square also tells us that the vertex of y = x² + 6x + 2 is at (−3, −7).

两边开平方得 x + 3 = ±√7，所以两根为 x = −3 + √7 和 x = −3 − √7。注意两根是无理数；配方还告诉我们 y = x² + 6x + 2 的顶点坐标为 (−3, −7)。

5. The Quadratic Formula | 求根公式

For any quadratic ax² + bx + c = 0, the roots are given by the quadratic formula. This method works for every quadratic equation, including those that cannot be factorised over the rational numbers. It is the most reliable method when you are running out of time or when the coefficients are awkward decimals.

对于任意一元二次方程 ax² + bx + c = 0，根由求根公式给出。该方法适用于一切二次方程，包括无法在有理数范围内因式分解的情形。当你时间紧张或系数是麻烦的小数时，它是最可靠的方法。

x = (−b ± √(b² − 4ac)) / (2a)

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Solving Quadratic Equations | 解一元二次方程