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 algebra topics in IGCSE Mathematics. They appear in algebraic manipulation, graph sketching, and real-world problems. Mastering the methods of solving them will give you confidence in both Core and Extended papers.

二次方程是 IGCSE 数学代数中最重要的主题之一。它们会出现在代数运算、图像绘制和实际应用问题中。掌握二次方程的解法能让你在 Core 和 Extended 试卷中更有把握。

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

A quadratic equation in one variable is an equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

一元二次方程是可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。

The expression ax² is called the quadratic term, bx is the linear term, and c is the constant term. If a = 0, the equation becomes linear, not quadratic.

ax² 称为二次项，bx 称为一次项，c 称为常数项。如果 a = 0，方程就变成了一次方程，不再是二次方程。

2. Solving by Factorising | 因式分解法

When a quadratic expression can be factorised into two linear factors, we can solve the equation quickly. Write the expression as (x – p)(x – q) = 0 and then set each bracket equal to zero.

当二次式可以分解为两个一次因式时，我们可以快速求解。把式子写成 (x – p)(x – q) = 0，然后令每个括号等于零。

Example: Solve x² – 7x + 12 = 0. Factorise to (x – 3)(x – 4) = 0, so x = 3 or x = 4.

例如：解 x² – 7x + 12 = 0。因式分解为 (x – 3)(x – 4) = 0，因此 x = 3 或 x = 4。

Always check that the factorised form expands back to the original expression.

一定要检查因式分解后的式子展开后是否能还原为原式。

3. The Zero Product Property | 零乘积性质

The zero product property states that if the product of two

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