Solving Quadratic Equations for IGCSE Maths | IGCSE 数学：解二次方程

📚 Solving Quadratic Equations for IGCSE Maths | IGCSE 数学：解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, and real-world problems, and examiners often test both the method and the interpretation of solutions. This article covers the standard form, three main solving techniques, the discriminant, graphical interpretation, and common mistakes.

二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、图像和实际问题中，考官经常同时考查解题方法和对方程解的解释。本文涵盖标准形式、三种主要解法、判别式、图像解释以及常见错误。

1. Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic. For example, 3x² − 5x + 2 = 0 has a = 3, b = −5, and c = 2.

二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程。例如，3x² − 5x + 2 = 0 中 a = 3、b = −5、c = 2。

Before solving, always rearrange the equation so that one side equals zero. Terms must be collected in descending powers of x. For example, x² + 4 = 5x should be rewritten as x² − 5x + 4 = 0.

求解前，一定要先把方程整理成一边等于零的形式。各项必须按 x 的降幂排列。例如，x² + 4 = 5x 应改写为 x² − 5x + 4 = 0。

2. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic expression can be factorised into two linear brackets. For x² − 5x + 6 = 0, we look for two numbers that multiply to +6 and add to −5: they are −2 and −3. So (x − 2)(x − 3) = 0.

当二次表达式可以因式分解为两个一次括号时，因式分解法通常是最快的方法。对于 x² − 5x + 6 = 0，我们寻找两个数，它们相乘得 +6，相加得 −5：这两个数是 −2 和 −3。因此 (x − 2)(x − 3) = 0。

Using the zero product property, if (x − 2)(x − 3) = 0, then x − 2 = 0 or x − 3 = 0. The solutions are x = 2 and x = 3. Always check each solution by substituting it back into the original equation.

根据零乘积性质，如果 (x − 2)(x − 3) = 0，那么 x − 2 = 0 或 x − 3 = 0。解为 x = 2 和 x = 3。每个解都要代回原方程检验。

When the coefficient of x² is not 1, such as 2x² + 7x + 3 = 0, factorising may require splitting the middle term or trial and error. The correct factorisation is (2x + 1)(x + 3) = 0, giving x = −1/2 or x = −3.

当 x² 的系数不是 1 时，例如 2x² + 7x + 3 = 0，因式分解可能需要拆分中间项或尝试。正确的因式分解是 (2x + 1)(x + 3) = 0，得到 x = −1/2 或 x = −3。

2x² + 7x + 3 = (2x + 1)(x + 3) = 0

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x − h)² + k = 0. This method is especially useful when the quadratic does not factorise neatly and when finding the vertex of a parabola.

配方法把二次方程改写为 a(x − h)² + k = 0 的形式。当二次方程不容易因式分解以及需要求抛物线的顶点时，这种方法特别有用。

For x² + 6x + 2 = 0, first write x² + 6x as (x + 3)² − 9. So the equation becomes (x + 3)² − 9 + 2 = 0, which simplifies to (x + 3)² = 7.

对于 x² + 6x + 2 = 0，先把 x² + 6x 写成 (x + 3)² − 9。于是方程变为 (x + 3)² − 9 + 2 = 0，简化为 (x + 3)² = 7。

Taking square roots gives x + 3 = ±√7, so x = −3 ± √7. The exact solutions are x = −3 + √7 and x = −3 − √7.

开平方得到 x + 3 = ±√7，因此 x = −3 ± √7。精确解为 x = −3 + √7 和 x = −3 − √7。

If the coefficient of x² is not 1, factor it out first. For example, 2x² + 8x + 3 = 0 becomes 2(x² + 4x) + 3 = 0. Complete the square inside the bracket: 2[(x + 2)² − 4] + 3 = 0, which leads to 2(x +

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