Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most heavily tested topics in IGCSE Mathematics, appearing in both Paper 2 and Paper 4. This revision guide covers every method you need — factorisation, completing the square, the quadratic formula, the discriminant, and applications — with worked examples and examiner tips throughout.

二次方程是 IGCSE 数学中最常考查的内容之一，在 Paper 2 和 Paper 4 中都会出现。本复习指南涵盖所有必备解法——因式分解法、配方法、二次公式、判别式及应用题——全程配有例题和考官提示。

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

A quadratic equation is an equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.

二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为实数，且 a ≠ 0。系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。

ax² + bx + c = 0, a ≠ 0

The highest power of the variable is 2, which is why the equation is called “quadratic.” A quadratic equation always has two solutions (called roots), but the two roots may be distinct, equal, or non-real. At IGCSE level you are expected to find real roots only.

变量的最高次数为 2，因此称为”二次”方程。二次方程总有两个解（称为根），但这两个根可能是相异的、相等的或无实根。在 IGCSE 阶段，只要求找到实根。

Examples of quadratic equations include x² − 5x + 6 = 0, 2x² + 3x − 7 = 0, and x² = 9. Notice that x² = 9 can be rewritten as x² − 9 = 0, so it is also in standard form.

二次方程的例子包括 x² − 5x + 6 = 0、2x² + 3x − 7 = 0 和 x² = 9。注意 x² = 9 可改写为 x² − 9 = 0，因此它也属于标准形式。

2. Method 1: Solving by Factorisation | 方法一：因式分解法

Factorisation is the quickest method when the quadratic expression can be written as the product of two linear brackets. The key principle is the zero product rule: if A × B = 0, then either A = 0 or B = 0.

当二次式可以写成两个线性括号相乘时，因式分解是最快的解法。关键原理是零积规则：若 A × B = 0，则 A = 0 或 B = 0。

If (px + q)(rx + s) = 0, then px + q = 0 or rx + s = 0.

Worked example 1: Solve x² − 5x + 6 = 0.

例题 1：解方程 x² − 5x + 6 = 0。

Step 1: Find two numbers that multiply to give the constant term +6 and add to give the coefficient of x, which is −5. These numbers are −2 and −3.

步骤 1：寻找两个数，使其乘积等于常数项 +6，和等于 x 的系数 −5。这两个数是 −2 和 −3。

Step 2: Write the factorised form: (x − 2)(x − 3) = 0.

步骤 2：写出因式分解形式：(x − 2)(x − 3) = 0。

Step 3: Set each bracket to zero: x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

步骤 3：令每个括号等于零：x − 2 = 0 或 x − 3 = 0，因此 x = 2 或 x = 3。

Worked example 2: Solve 3x² − 8x + 4 = 0. When a ≠ 1, multiply a and c: 3 × 4 = 12. Find two numbers whose product is 12 and sum is −8: those are −2 and −6. Rewrite the middle term: 3x² − 2x − 6x + 4 = 0. Factor by grouping: x(3x − 2) − 2(3x − 2) = 0, giving (3x − 2)(x − 2) = 0. Hence x = 2/3 or x = 2.

例题 2：解方程 3x² − 8x + 4 = 0。当 a ≠ 1 时，将 a 与 c 相乘：3 × 4 = 12。寻找两个数，积为 12，和为 −8：这两个数是 −2 和 −6。改写中间项：3x² − 2x − 6x + 4 = 0。分组因式分解：x(3x − 2) − 2(3x − 2) = 0，得 (3x − 2)(x − 2) = 0。故 x = 2/3 或 x = 2。

Always check whether there is a common factor to remove before attempting to factorise further. For example, 2x² + 8x + 6 = 0 should first be divided by 2 to give x² + 4x + 3 = 0.

在进一步因式分解之前，务必先检查是否有公因数可以提取。例如，2x² + 8x + 6 = 0 应先除以 2，得到 x² + 4x + 3 = 0。

3. Method 2: Completing the Square | 方法二：配方法

Completing the square rewrites a quadratic in the form a(x + h)² + k. This method always works, even when the expression cannot be factorised easily, and it also reveals the vertex of the parabola.

配方法将二次式改写为 a(x + h)² + k 的形式。即使表达式不易因式分解，此方法也始终有效，并且还能揭示抛物线的顶点。

x² + bx = (x + b/2)² − (b/2)²

Worked example: Solve x² + 8x + 5 = 0 by completing the square.

例题：用配方法解方程 x² + 8x + 5 = 0。

Step 1: Move the constant term to the right: x² + 8x = −5.

步骤 1：将常数项移到右边：x² + 8x = −5。

Step 2: Take half of 8, which is 4, and square it to get 16. Add 16 to both sides: x² + 8x + 16 = −5 + 16.

步骤 2：取 8 的一半为 4，其平方为 16。两边同时加 16：x² + 8x + 16 = −5 + 16。

更多咨询请联系16621398022（同微信）