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. They appear in algebra, geometry, graphs, and even calculus preparation. Mastering the different methods of solving quadratics — factorisation, the quadratic formula, and completing the square — is essential for exam success.

二次方程是 IGCSE 数学中考查最多的知识点之一。它出现在代数、几何、函数图象甚至为微积分奠基的内容中。掌握因式分解法、二次公式和配方法这三大解法，是考试取得好成绩的关键。

1. The Standard Form | 标准形式

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the unknown x is 2, which is why it is called “quadratic” (from the Latin “quadratus” meaning square).

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。未知数 x 的最高次数为 2，因此被称为”二次”（quadratic 源自拉丁语，意为”平方”）。

For example, 2x² − 3x + 1 = 0 is a quadratic equation, while x³ − 2x = 0 is not. A quadratic equation can have two, one, or zero real solutions — these solutions are called the roots or zeros of the equation.

例如，2x² − 3x + 1 = 0 是二次方程，而 x³ − 2x = 0 不是。二次方程可以有两个、一个或零个实数解——这些解被称为方程的根或零点。

2. Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic has “nice” integer coefficients. To factorise x² + bx + c, find two numbers that multiply to give c and add to give b.

因式分解通常是处理整数系数时最快的方法。要将 x² + bx + c 分解因式，需要找到两个数，使它们的乘积等于 c，和等于 b。

For the equation x² − 5x + 6 = 0, we need two numbers whose product is 6 and whose sum is −5. These are −2 and −3. So the equation becomes (x − 2)(x − 3) = 0. If a product is zero, then at least one factor is zero, giving x = 2 or x = 3.

对方程 x² − 5x + 6 = 0，我们需要找两个数，乘积为 6，和为 −5。它们是 −2 和 −3。于是方程变为 (x − 2)(x − 3) = 0。由于乘积为零时至少有一个因式为零，所以得到 x = 2 或 x = 3。

When a ≠ 1, for example 2x² + 7x + 3 = 0, multiply a and c (2 × 3 = 6). Find two numbers that multiply to 6 and add to 7: these are 1 and 6. Then split the middle term and factor by grouping.

当 a ≠ 1 时，例如 2x² + 7x + 3 = 0，先将 a 与 c 相乘（2 × 3 = 6）。找两个数，乘积为 6 且和为 7：它们是 1 和 6。然后拆项分组分解因式。

2x² + 7x + 3 = 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

Therefore x = −½ or x = −3. Remember: if the equation is not given in the form ax² + bx + c = 0, rearrange it first.

因此 x = −½ 或 x = −3。注意：如果方程不是 ax² + bx + c = 0 的形式，要先整理为这种形式。

3. The Quadratic Formula | 二次公式

Not every quadratic can be factorised easily. The quadratic formula works for every quadratic equation and is derived from completing the square. Given ax² + bx + c = 0, the roots are given by:

并非所有二次方程都能轻易分解因式。二次公式适用于一切二次方程，它由配方法推导而来。对于 ax² + bx + c = 0，其根为：

x = (−b ± √(b² − 4ac)) / 2a

This formula is provided in many IGCSE formula booklets, but you must know exactly when and how to apply it. Always substitute a, b, c carefully and simplify the surd if possible.

许多 IGCSE 公式册中会提供该公式，但你必须

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