Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, coordinate geometry, functions, and even in problem-solving questions that model real-life situations. Understanding how to solve quadratic equations efficiently and accurately is therefore a key skill for every student aiming for a high grade.

二次方程是IGCSE数学中最重要的主题之一。它出现在代数、坐标几何、函数，甚至模拟现实生活情境的应用题中。因此，高效准确地解二次方程是每个追求高分的学生的关键技能。

1. Standard Form | 标准形式

Every quadratic equation can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a is not equal to zero. The coefficient a defines the curvature of the parabola: if a > 0 the curve opens upward, and if a < 0 it opens downward. When a = 0, the equation becomes linear, and many of the techniques explained below no longer apply.

每个二次方程都可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数，且 a 不等于零。系数 a 决定了抛物线的开口方向：如果 a > 0，曲线开口向上；如果 a < 0，曲线开口向下。当 a = 0 时，方程变为线性方程，下文中的许多方法不再适用。

ax² + bx + c = 0 (a ≠ 0)

2. Factorisation Method | 因式分解法

Factorisation is the fastest method when the quadratic expression can be written as the product of two linear factors. The approach relies on the zero product property: if two numbers multiply to give zero, then at least one of them must be zero. For example, from (x − 2)(x − 5) = 0 we immediately obtain x = 2 or x = 5.

当二次表达式可以写成两个线性因子的乘积时，因式分解是最快的方法。该方法依赖零乘积性质：如果两个数相乘为零，则其中至少一个必须为零。例如，从 (x − 2)(x − 5) = 0 我们可以直接得到 x = 2 或 x = 5。

• Move all terms to one side so the right-hand side equals zero. | 将所有项移到一边，使右边等于零。
• Factorise the expression into two brackets. | 将左边分解成两个括号。
• Set each bracket equal to zero. | 令每个括号等于零。
• Solve the two linear equations. | 解出两个线性方程。

Example: x² − 7x + 10 = 0 can be written as (x − 2)(x − 5) = 0, so x = 2 or x = 5.

例子：x² − 7x + 10 = 0 可以写成 (x − 2)(x − 5) = 0，因此 x = 2 或 x = 5。

3. Quadratic Formula | 二次公式

When factorisation is not obvious or the roots are irrational, the quadratic formula provides a universal method. By substituting the coefficients a, b and c into the formula below, we can always find the roots, whether they are rational or irrational.

当因式分解不明显或根为无理数时，二次公式提供了通用解法。将系数 a、b、c 代入下面的公式，我们总能求出根，无论它们是有理数还是无理数。

x = (−b ± √(b² − 4ac)) / 2a

Example: Solve 2x² + 3x − 2 = 0 using the quadratic formula. Here a = 2, b = 3 and c = −2.

例子：用二次公式解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

x = (−3 ± √(9 − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √25) / 4 = (−3 ± 5) / 4

Therefore x = 1/2 or x = −2. Always rearrange the equation into standard form before substituting a, b and c.

因此 x = 1/2 或 x =

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