Mastering Quadratic Equations | 掌握一元二次方程

📚 Mastering Quadratic Equations | 掌握一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, geometry and problem-solving questions, and mastering them can earn you many marks across several papers.

一元二次方程是 IGCSE 数学最重要的考点之一。它贯穿代数、图像、几何与应用题，掌握好这一主题能在多份试卷中为你赢取大量分数。

1. Understanding Quadratic Equations | 理解一元二次方程

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the unknown x is 2. Its standard form is:

一元二次方程是最高次数为 2 的多项式方程，即未知数 x 的最高幂次为 2。其标准形式为：

ax² + bx + c = 0, a ≠ 0

Here a, b and c are constants, and a must not be zero. If a = 0, the term ax² disappears and the equation becomes linear, not quadratic.

其中 a、b、c 为常数，且 a 不能为零。若 a = 0，则 ax² 项消失，方程退化为一次方程，而不再是二次方程。

For example:

例如：

• x² − 6x + 8 = 0 is quadratic because a = 1, b = −6, c = 8.
• x² − 6x + 8 = 0 是二次方程，因为 a = 1，b = −6，c = 8。

• 2x² + 5x − 3 = 0 is quadratic because a = 2, b = 5, c = −3.
• 2x² + 5x − 3 = 0 是二次方程，因为 a = 2，b = 5，c = −3。

• 3x + 1 = 0 is not quadratic because there is no x² term.
• 3x + 1 = 0 不是二次方程，因为其中没有 x² 项。

2. Expanding and Factorising Quadratics | 展开与因式分解

To solve quadratic equations efficiently, you must be confident with factorising. Factorising is the reverse process of expanding brackets.

要高效地解一元二次方程，你需要熟练掌握因式分解。因式分解是展开括号的逆运算。

For a monic quadratic x² + bx + c, look for two numbers p and q such that p + q = b and pq = c. Then:

对于首项系数为 1 的二次式 x² + bx + c，找出两个数 p、q，使 p + q = b，pq = c。于是：

x² + bx + c = (x + p)(x + q)

Example: Factorise x² − 6x + 8. We need p + q = −6 and pq = 8. The numbers are −2 and −4, so x² − 6x + 8 = (x − 2)(x − 4).

例如：分解 x² − 6x + 8，需要 p + q = −6，pq = 8，取 −2 和 −4，因此 x² − 6x + 8 = (x − 2)(x − 4)。

Another important pattern is the difference of two squares:

另一个重要的公式是平方差公式：

a² − b² = (a − b)(a + b)

For example, x² − 25 = (x − 5)(x + 5), and 4x² − 9 = (2x − 3)(2x + 3).

例如：x² − 25 = (x − 5)(x + 5)，4x² − 9 = (2x − 3)(2x + 3)。

Always check whether a common factor can be taken out first, such as 2x² + 6x = 2x(x + 3).

还要注意先提取公因式，例如 2x² + 6x = 2x(x + 3)。

3. Solving by Factorisation | 用因式分解法求解

The factorisation method uses the zero product property: if the product of two expressions is zero, then at least one of the expressions must be zero.

因式分解法依据零因子性质：若两个因

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