Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程精讲

📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程精讲

Quadratic equations are one of the most important algebra topics in IGCSE Mathematics. They appear in Paper 2 and Paper 4, both as standalone questions and inside larger problems such as area, projectile motion, and optimisation. This article explains the key methods you need to master: factorising, completing the square, and using the quadratic formula.

二次方程是 IGCSE 数学代数部分最重要的主题之一。它们在 Paper 2 和 Paper 4 中都会出现，既可能独立出题，也可能出现在面积、抛体运动和优化等大型问题中。本文将讲解你需要掌握的关键方法：因式分解法、配方法和求根公式法。

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

A quadratic equation is an equation in which the highest power of the variable is 2. Its general shape is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是指未知数的最高次数为 2 的方程。它的一般形式为 ax² + bx + c = 0，其中 a、b 和 c 是常数，且 a ≠ 0。

For example, 3x² − 5x + 2 = 0 is quadratic because the term 3x² has degree 2. The equation x³ + x = 0 is not quadratic because the highest power is 3.

例如，3x² − 5x + 2 = 0 是二次方程，因为 3x² 这一项的次数为 2。方程 x³ + x = 0 不是二次方程，因为最高次数是 3。

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

Before solving, always rearrange the equation into standard form ax² + bx + c = 0. The coefficient a is the number in front of x², b is in front of x, and c is the constant term.

求解之前，一定要先把方程整理成标准形式 ax² + bx + c = 0。系数 a 是 x² 前面的数，b 是 x 前面的数，c 是常数项。

• 2x² + 3x − 5 = 0 has a = 2, b = 3, c = −5.
• 2x² + 3x − 5 = 0 中 a = 2，b = 3，c = −5。

If the equation is given as 5 − 4x = 3x², move all terms to one side: 3x² + 4x − 5 = 0. Keeping the standard form prevents sign errors when using formulas.

如果方程给出的是 5 − 4x = 3x²，需要把所有项移到一边：3x² + 4x − 5 = 0。保持标准形式可以避免在使用公式时出现符号错误。

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3. Solving by Factorising | 因式分解法

Factorising is usually the fastest method when the quadratic has simple integer roots. Write the quadratic as a product of two brackets: (px + q)(rx + s) = 0.

当二次方程有简单的整数根时，因式分解通常是最快的方法。把二次式写成两个括号的乘积：(px + q)(rx + s) = 0。

For x² − 5x + 6 = 0, find two numbers that multiply to +6 and add to −5. These are −2 and −3, so x² − 5x + 6 = (x − 2)(x − 3) = 0.

对于 x² − 5x + 6 = 0，找到两个数，它们相乘为 +6、相加为 −5。这两个数是 −2 和 −3，因此 x² − 5x + 6 = (x − 2)(x − 3) = 0。

For non-monic quadratics such as 2x² + 7x + 3, find two numbers that multiply to a × c = 6 and add to b = 7. The numbers are 6 and 1, so rewrite 7x as 6x + x and factor by grouping: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

对于非首一二次式，如 2x² + 7x + 3，先找到两个数，它们相乘为 a × c = 6、相加为 b = 7。这两个数是 6 和 1，于是把 7x 拆成 6x + x，再分组分解：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

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4. Zero Product Property | 零乘积性质

After factorising, the equation becomes (px + q)(rx + s) = 0. The zero product property tells us that if two factors multiply to zero, at least one factor must be zero.

因式分解后，方程变为 (px + q)(rx + s) = 0。零乘积性质告诉我们，如果两个因式的乘积为零，那么至少有一个因式必须为零。

Set each bracket equal to zero and solve separately. For (x − 2)(x − 3) = 0, either x − 2 = 0, giving x = 2, or x − 3 = 0, giving x = 3.

令每个括号分别等于零并求解。对于 (x − 2)(x − 3) = 0，要么 x − 2 = 0，解得 x = 2；要么 x − 3 = 0，解得 x = 3。

This property only works when one side of the equation is exactly zero. If you have (x − 2)(x − 3) = 4, do not set each bracket equal to 4. You must expand, rearrange to zero, and then factorise again.

这一性质只在方程一边恰好等于零时才能使用。如果你遇到 (x − 2)(x − 3) = 4，不要把每个括号分别设为 4。你必须展开、移项使左边等于零，然后再重新因式分解。

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5. Completing the Square | 配方法

Completing the square is a powerful technique because it works for every quadratic, even when factorising is difficult. It is also used to find the turning point of a quadratic graph.

配方法是一种非常有效的方法，因为它适用于每一个二次方程，即使在因式分解很困难时也能使用。它还可用于求二次函数图像的顶点。

Start with x² + bx + c = 0. Halve the coefficient of x, square it, and add and subtract that value inside the expression:

从 x² + bx + c = 0 开始。把 x 的系数除以 2，再平方，然后在表达式中同时加上和减去这个值：

x² + bx + c = (x + b/2)² − (b/2)² + c

Example: x² + 6x + 5 = 0. Half of 6 is 3, and 3² = 9, so x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4 = 0.

例如：x² + 6x + 5 = 0。6 的一半是 3，3² =

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