Quadratic Equations: Factorising, Completing the Square & the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square & the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

Quadratic equations form one of the most important algebra topics in IGCSE Mathematics. They appear in word problems, graphs, geometry and many real-life models. This article explains the main solving techniques – factorising, completing the square and the quadratic formula – with worked examples and exam tips.

二次方程是 IGCSE 数学中最重要的代数主题之一。它们出现在应用题、图像、几何以及许多实际模型中。本文将解释主要的求解方法——因式分解、配方法和求根公式，并提供例题与考试技巧。

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

A quadratic equation is a polynomial equation of degree two. It can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were zero, the equation would become linear. Quadratic equations appear throughout the IGCSE syllabus, from area problems to projectile motion.

二次方程是一个二次多项式方程。它的一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。如果 a 为 0，方程就变成一次方程。二次方程在 IGCSE 大纲中随处可见，从面积问题到抛体运动都会用到。

The word ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square. It refers to the x² term, which is the highest power in the equation. A quadratic equation usually has two solutions, called roots, although sometimes the two roots are the same number.

“二次”一词来源于拉丁语 ‘quadratus’，意为正方形。它指的是方程中最高次项 x²。二次方程通常有两个解，称为根，尽管有时两个根是同一个数。

2. Standard Form and Key Terms | 标准形式与关键术语

Before solving, always rewrite the equation in standard form ax² + bx + c = 0. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. Terms should be collected on one side so the other side equals zero.

在求解之前，必须先把方程整理成标准形式 ax² + bx + c = 0。系数 a 是二次项系数，b 是一次项系数，c 是常数项。所有项要移到同一边，使另一边等于零。

For example, x² + 3x = 4 is not yet in standard form. Subtracting 4 from both sides gives x² + 3x − 4 = 0. In this case, a = 1, b = 3 and c = −4. Getting the standard form right prevents sign errors later.

例如，x² + 3x = 4 还不是标准形式。两边同时减去 4，得到 x² + 3x − 4 = 0。此时 a = 1，b = 3，c = −4。正确整理成标准形式可以避免后续的符号错误。

You may also meet the words ‘root’ and ‘solution’. They mean the same thing: the values of x that make the equation true. A root satisfies the equation, so substituting it back gives 0 = 0.

你还会遇到“根”和“解”这两个词。它们意思相同：使方程成立的 x 值。根满足方程，因此将它代回原方程会得到 0 = 0。

3. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic can be written as a product of two linear brackets. If pq = 0, then p = 0 or q = 0. This zero-product property is the key step in solving factorised quadratics.

当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。如果 pq = 0，那么 p = 0 或 q = 0。这个“零积性质”是求解因式分解后的二次方程的关键步骤。

For a monic quadratic x² + bx + c, find two numbers that multiply to c and add to b. Then write x² + bx + c = (x + m)(x + n). If the two numbers are negative, the signs inside the brackets must be adjusted accordingly.

对于首项系数为 1 的二次式 x² + bx + c，找到两个数，使它们的乘积等于 c，和等于 b。然后写成 x² + bx + c = (x + m)(x + n)。如果这两个数是负数，括号内的符号要相应调整。

For non-monic quadratics such as 2x² + 7x + 3, use either trial and error or the ac method: multiply a by c, find factor pairs, split the middle term, and factor by grouping.

对于非首一二次式，如 2x² + 7x + 3，可以用试错法或 ac 法：先将 a 与 c 相乘，寻找因数对，拆分中间项，再分组分解。

4. Worked Example: Factorising | 因式分解例题

Solve x² + 5x + 6 = 0. We need two numbers whose product is 6 and sum is 5. The numbers are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3).

解方程 x² + 5x + 6 = 0。我们需要两个数，乘积为 6，和为 5。这两个数是 2 和 3，所以 x² + 5x + 6 = (x + 2)(x + 3)。

Set each bracket equal to zero: x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3. The solutions are x = −2 or x = −3. Check by substitution: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0, and (−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0.

令每个括号等于零：x + 2 = 0 得 x = −2；x + 3 = 0 得 x = −3。方程的解为 x = −2 或 x = −3。代入检验：(−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0；(−3)² + 5(−3) + 6 = 9 − 15 + 6 = 0。两者

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