Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is a polynomial equation of degree 2, which means the highest power of the variable is 2. In IGCSE Mathematics, solving quadratic equations is a core skill that appears in almost every examination paper.

二次方程是最高次数为2的多项式方程。在IGCSE数学中，解二次方程是几乎每份考卷都会出现的核心技能。

1. Standard Form and Terminology | 标准形式与术语

Every quadratic equation can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The value of a cannot be zero because if a = 0, the equation becomes linear, not quadratic.

每个二次方程都可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。a 不能为零，因为若 a = 0，方程就变成了一次方程而非二次方程。

The three terms have specific names. ax² is the quadratic term, bx is the linear term, and c is the constant term. For example, in 2x² − 5x + 3 = 0, we have a = 2, b = −5 and c = 3.

这三个项有各自的名称：ax² 是二次项，bx 是一次项，c 是常数项。例如，在 2x² − 5x + 3 = 0 中，a = 2、b = −5、c = 3。

Solutions to a quadratic equation are also called roots or zeros. A quadratic equation can have two real roots, one repeated root, or no real roots, depending on the value of the discriminant, which we will explore later in this article.

二次方程的解也称为根或零点。二次方程可以有两个实根、一个重根，或者没有实根——这取决于判别式的值，我们将在后文详细讨论。

2. Solving by Factorisation | 用因式分解法解二次方程

Factorisation is often the quickest method when the quadratic has simple integer coefficients. The method relies on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

当二次方程具有简单的整数系数时，因式分解通常是最快捷的方法。该方法依赖于零积性质：如果两个表达式的乘积为零，那么至少其中一个必须为零。

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

例题1：解方程 x² + 5x + 6 = 0。

We need two numbers that multiply to give 6 and add to give 5. The numbers are 2 and 3, because 2 × 3 = 6 and 2 + 3 = 5. Therefore x² + 5x + 6 = (x + 2)(x + 3).

我们需要找到两个数，它们的乘积为6、和为5。这两个数是2和3，因为 2 × 3 = 6 且 2 + 3 = 5。因此 x² + 5x + 6 = (x + 2)(x + 3)。

Setting each factor to zero: x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3. The solution set is x = −2 or x = −3.

令每个因式为0：x + 2 = 0 得 x = −2；x + 3 = 0 得 x = −3。解集为 x = −2 或 x = −3。

Worked Example 2: Solve 2x² − 7x + 3 = 0.

例题2：解方程 2x² − 7x + 3 = 0。

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