Mastering Quadratic Equations for IGCSE | 精通 IGCSE 二次方程

📚 Mastering Quadratic Equations for IGCSE | 精通 IGCSE 二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus, appearing in algebra, graphs, problem-solving and even geometry questions. This revision guide explains how to recognise, solve and apply quadratics using factorisation, completing the square, the quadratic formula and graphical methods. Work through each section carefully and practise the examples to build confidence for your exam.

二次方程是 IGCSE 数学大纲中最重要的主题之一，出现在代数、图像、问题求解甚至几何题中。本复习指南讲解如何识别、求解和应用二次方程，包括因式分解法、配方法、二次公式法和图像法。请仔细学习每一节并练习例题，为考试建立信心。

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

A quadratic equation is a polynomial equation in which the highest power of the variable is 2. It can always be written in the general form ax² + bx + c = 0, where a, b and c are constants and a is not equal to zero. If a = 0, the equation becomes linear, not quadratic.

二次方程是指未知数的最高次数为 2 的多项式方程。它总是可以写成一般形式 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不等于零。如果 a = 0，方程就变成一次方程，不再是二次方程。

For example, x² − 5x + 6 = 0 and 3x² + 2x − 1 = 0 are quadratic equations. The equation 4x + 7 = 0 is not quadratic because the highest power of x is 1.

例如，x² − 5x + 6 = 0 和 3x² + 2x − 1 = 0 都是二次方程。方程 4x + 7 = 0 不是二次方程，因为 x 的最高次数是 1。

2. Standard Form and Coefficients | 标准形式与系数

The standard form of a quadratic equation is ax² + bx + c = 0. The number a is called the leading coefficient, b is the linear coefficient, and c is the constant term. It is essential to rearrange an equation into this form before solving it by factorisation or by the quadratic formula.

二次方程的标准形式是 ax² + bx + c = 0。数字 a 称为二次项系数，b 称为一次项系数，c 称为常数项。在用因式分解法或二次公式法求解之前，必须先将方程整理成这种形式。

For instance, the equation 2x² = 5x + 3 should be rearranged as 2x² − 5x − 3 = 0. Here a = 2, b = −5 and c = −3. Keeping signs with each term is important because a mistake in sign changes the solutions.

例如，方程 2x² = 5x + 3 应整理为 2x² − 5x − 3 = 0。此时 a = 2，b = −5，c = −3。保留每一项的符号非常重要，因为符号错误会改变解。

• a is the coefficient of x²; it cannot be zero.
• b is the coefficient of x.
• c is the constant term.

• a 是 x² 的系数，不能为零。
• b 是 x 的系数。
• c 是常数项。

3. Solving by Factorisation | 因式分解法

Factorisation is often the fastest way to solve a quadratic equation when the expression can be written as a product of two linear factors. The method uses the zero product property: if the product of two factors is zero, then at least one of the factors must be zero.

当二次表达式可以写成两个一次因式的乘积时，因式分解法通常是最快的求解方法。该方法利用零乘积性质：如果两个因式的乘积为零，那么至少有一个因式必须为零。

Procedure: first write the equation in standard form. Then factorise the left-hand side. Set each factor equal to zero and solve the resulting linear equations. Always check your answers by substituting them back into the original equation.

步骤：首先将方程写成标准形式。然后将左边进行因式分解。令每个因式等于零并求解得到的一次方程。一定要将答案代回原方程进行检验。

Example: Solve x² − 7x + 12 = 0. Factorise as (x − 3)(x − 4) = 0. Set x − 3 = 0 or x − 4 = 0. Therefore x = 3 or x = 4.

例题：解 x² − 7x + 12 = 0。因式分解为 (x − 3)(x − 4) = 0。令 x − 3 = 0 或 x − 4 = 0。因此 x = 3 或 x = 4。

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