Mastering Quadratic Equations for IGCSE | IGCSE 二次方程高分突破

📚 Mastering Quadratic Equations for IGCSE | IGCSE 二次方程高分突破

A quadratic equation is one of the most important topics in the IGCSE Mathematics syllabus. It appears in algebra papers, graph-sketching questions, and real-life word problems, so a clear method is essential for top marks.

二次方程是 IGCSE 数学大纲中最重要的主题之一。它们出现在代数试卷、函数图像题和现实生活应用题中，因此掌握清晰的方法对取得高分至关重要。

1. What Quadratic Equations Look Like | 二次方程长什么样

A quadratic equation in one variable 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. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.

一个一元二次方程是任何能写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。ax² 叫做二次项，bx 叫做一次项，c 叫做常数项。

If a were equal to zero, the equation would become bx + c = 0, which is only linear. This is why the condition a ≠ 0 is essential at IGCSE. Examiners often include questions where students must identify whether an equation is truly quadratic.

如果 a 等于 0，方程就会变成 bx + c = 0，这只是一次方程。所以 a ≠ 0 这个条件在 IGCSE 考试中非常重要。考官经常出题考查学生能否判断一个方程是否真的是二次方程。

ax² + bx + c = 0, a ≠ 0

2. Solving by Factorising | 因式分解法

When the quadratic expression can be factorised, this method is usually the fastest. For example, consider x² + 7x + 12 = 0. We look for two numbers that multiply to 12 and add to 7. These are 3 and 4, so the factorised form is (x + 3)(x + 4) = 0.

当二次式可以因式分解时，这个方法通常最快。例如 x² + 7x + 12 = 0。我们要找两个数，乘积为 12，和为 7。它们是 3 和 4，因此因式分解形式为 (x + 3)(x + 4) = 0。

Then apply the null factor law: either x + 3 = 0 or x + 4 = 0, giving x = −3 or x = −4. Always check your answers by substituting them back into the original equation.

然后使用零因子定律：要么 x + 3 = 0，要么 x + 4 = 0，得到 x = −3 或 x = −4。一定要把答案代回原方程进行检验。

For equations such as 2x² + 5x − 3 = 0, first find two numbers that multiply to 2 × (−3) = −6 and add to 5: these are 6 and −1. Split the middle term and factorise by grouping to obtain (2x − 1)(x + 3) = 0.

对于 2x² + 5x − 3 = 0 这类方程，先找两个数，乘积为 2 × (−3) = −6，和为 5：它们是 6 和 −1。拆分中间项，再用分组法分解，得到 (2x − 1)(x + 3) = 0。

3. The Null Factor Law | 零因子定律

The null factor law states that if A × B = 0, then A = 0 or B = 0. This is the reason factorising is so powerful. Once a quadratic is written as a product of two brackets equal to zero, each bracket can be set to zero separately.

零因子定律指出：如果 A × B = 0，那么 A = 0 或 B = 0。这就是因式分解法特别有效的原因。一旦二次方程写成两个括号的乘积等于 0，就可以分别令每个括号等于 0。

It only works when the product equals zero. If (x − 2)(x + 5) = 3, you cannot simply set x − 2 = 3 or x + 5 = 3. You must first expand, rearrange to standard form, and then factorise.

它只在乘积等于 0 时成立。如果 (x − 2)(x + 5) = 3，不能直接令 x − 2 = 3 或 x + 5 = 3。你必须先展开，整理成标准形式，再因式分解。

A common exam trap is dividing both sides by a variable. For example, x² = 5x should become x² − 5x = 0, then x(x − 5) = 0, giving x = 0 or x = 5. Dividing by x would lose the root x = 0.

常见的考试陷阱是两边同时除以一个变量。例如 x² = 5x 应该写成 x² − 5x = 0，然后 x(x − 5) = 0，得到 x = 0 或 x = 5。两边除以 x 会丢掉 x = 0 这个根。

4. Difference of Two Squares | 平方差公式

Some quadratics have no linear term, such as x² − 9 = 0. These can be solved using the difference of two squares: a² − b² = (a − b)(a + b). Recognising this pattern can save valuable time in the exam.

有些二次方程没有一次项，例如 x² − 9 = 0。这类方程可以用平方差公式 a² − b² = (a − b)(a + b) 来解。识别出这个模式可以在考试中节省宝贵的时间。

Write x² − 9 = 0 as (x − 3)(x + 3) = 0, so x = 3 or x = −3. This is much faster than using the quadratic formula or completing the square.

把 x² − 9 = 0 写成 (x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。这比套求根公式或配方法快得多。

For 4x² − 25 = 0, it becomes (2x − 5)(2x + 5) = 0, giving x = 5/2 or x = −5/2. Always check whether the two terms are perfect squares before choosing this method.

对于 4x² − 25 = 0，它变为 (2x − 5)(2x + 5) = 0，得到 x = 5/2 或 x = −5/2。在选择这个方法之前，一定要检查两项是否都是完全平方。

5. Completing the Square | 配

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