Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in nearly every exam paper, often in both the non-calculator and calculator sections. Mastering the techniques in this article will help you solve any quadratic equation with confidence.

二次方程是 IGCSE 数学中最重要的考点之一，几乎每份试卷都会出现，包括非计算器与计算器部分。掌握本文介绍的方法，你就能自信地解出任何二次方程。

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1. What Is a Quadratic Equation? | 什么是二次方程

A quadratic equation is a polynomial equation of degree 2. Its highest power of the variable is 2, which is why it is called “quadratic” (from the Latin word “quadratus”, meaning square). The general form is shown below, where a, b and c are constants and a ≠ 0.

二次方程是最高次数为 2 的多项式方程，变量的最高次幂是 2，因此称为 “quadratic”（源自拉丁语 “quadratus”，意为平方）。其一般形式如下，其中 a、b、c 为常数且 a ≠ 0。

ax² + bx + c = 0

The condition a ≠ 0 is essential: if a = 0, the equation becomes linear (bx + c = 0), not quadratic. In IGCSE, you will meet equations already in this form as well as those you must rearrange first.

条件 a ≠ 0 至关重要：若 a = 0，方程退化为线性方程（bx + c = 0），不再是二次方程。在 IGCSE 中，你会遇到已经符合标准形式的方程，也需要处理必须先移项整理的方程。

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2. Standard Form and Key Terms | 标准形式与关键术语

In the standard form ax² + bx + c = 0, the term ax² is called the quadratic term, bx is the linear term, and c is the constant term. The coefficient a is the leading coefficient. Always rewrite an equation into standard form before solving.

在标准形式 ax² + bx + c = 0 中，ax² 称为二次项，bx 称为一次项，c 称为常数项。系数 a 为首项系数。求解前一定要先将方程整理成标准形式。

For example, the equation 2x² = 5x − 3 must be rearranged: subtract 5x and add 3 to both sides to obtain 2x² − 5x + 3 = 0. Here a = 2, b = −5, c = 3.

例如，方程 2x² = 5x − 3 需要重新整理：两边同时减去 5x 并加上 3，得到 2x² − 5x + 3 = 0。此时 a = 2，b = −5，c = 3。

It is good practice to keep a positive, as this simplifies factorisation. If a is negative, multiply the whole equation by −1.

建议保持 a 为正数，这样因式分解会更简单。若 a 为负数，将整个方程乘以 −1 即可。

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3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer coefficients. The core principle is the zero product property: if the product of two factors is zero, then at least one of the factors must be zero.

当二次方程具有简单的整数系数时，因式分解是最快的方法。核心原理是零积性质：若两个因式的乘积为零，则至少有一个因式为零。

If AB = 0, then A = 0 or B = 0

Consider x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5 — those numbers are 2 and 3. Hence (x + 2)(x + 3) = 0. Setting each factor to zero gives x = −2 or x = −3.

考虑 x² + 5x + 6 = 0。我们需要找到两个数，相乘得 6 且相加得 5 —— 这两个数是 2 和 3。因此 (x + 2)(x + 3) = 0。令每个因式等于零，得 x = −2 或 x = −3。

For equations with a ≠ 1, such as 2x² + 7x + 3 = 0, multiply a and c (2 × 3 = 6) and find two numbers that multiply to 6 and add to 7. These are 6 and 1. Split the middle term: 2x² + 6x + x + 3 = 0, then factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. Hence x = −1/2 or x = −3.

对于 a ≠ 1 的方程，如 2x² + 7x + 3 = 0，将 a 与 c 相乘（2 × 3 = 6），找到两个数相乘得 6 且相加得 7。这两个数是 6 和 1。拆分中间项：2x² + 6x + x + 3 = 0，再分组因式分解：2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。因此 x = −1/2 或 x = −3。

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4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² = q, after which the equation can be solved by taking square roots. This method works for every quadratic equation.

配方法将二次方程改写为 (x + p)² = q 的形式，然后通过开平方求解。此方法适用于所有二次方程。

To complete the square for x² + bx + c = 0, take half of b, square it, then add and subtract this value.

对于 x² + bx + c = 0 配方时，取 b 的一半，将它的平方加减到方程中。

Example: solve x² + 6x + 2 = 0. Half of 6 is 3, and 3² = 9. Rewrite as x² + 6x + 9 − 9 + 2 = 0, giving (x + 3)² − 7 = 0. Then (x + 3)² = 7, so x + 3 = ±√7, hence x = −3 ± √7.

示例：解 x² + 6x + 2 = 0。6 的一半是 3，且 3² = 9。改写为 x² + 6x + 9 − 9 + 2 = 0，即 (x + 3)² − 7 = 0。于是 (x + 3)² = 7，所以 x + 3 = ±√7，得到 x = −3 ± √7。

If the coefficient of x² is not 1, divide the entire equation by a first, then complete the square.

若 x² 的系数不为 1，先将整个方程除以 a，再进行配方。

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5. The Quadratic Formula | 二次求根公式

The quadratic formula is a universal tool — it works for any quadratic equation, whether or not it factorises neatly. In IGCSE, this formula is provided on the formula sheet, but you must know how to substitute into it accurately using your calculator.

二次求根公式是通用工具，适用于任何二次方程，无论其能否整齐地因式分解。在 IGCSE 考试中，公式表会提供该公式，但你必须学会用计算器准确代入计算。

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, write the equation in standard form ax² + bx + c = 0, identify a, b and c, then substitute. The ± sign produces two solutions: one with addition and one with subtraction.

使用公式时，先将方程写成标准形式 ax² + bx + c = 0，确定 a、b、c，再代入。± 号产生两个解：一个相加、一个相减。

Example: solve 2x² + 3x − 5 = 0

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