Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most frequently tested topics in IGCSE Mathematics. Whether you sit for the Core or Extended paper, mastering quadratics unlocks marks across algebra, graph work and problem-solving. This revision guide breaks down every method you need, with step-by-step examples and exam-style advice.

二次方程是 IGCSE 数学中最常考的内容之一。无论你参加 Core 还是 Extended 试卷，掌握二次方程都能帮你解锁代数、函数图像和实际应用题中的分数。本复习指南系统梳理所有必备解法，辅以分步例题与应试建议。

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

A quadratic equation is a polynomial equation of degree 2. Its standard form is shown below.

二次方程是次数为 2 的多项式方程，其标准形式如下所示。

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants, and a cannot be zero. If a were zero, the equation would become linear (bx + c = 0). The term ax² is called the quadratic term, bx the linear term, and c the constant term.

其中 a、b、c 为常数，且 a 不能为 0。若 a 为 0，方程将退化为一次方程（bx + c = 0）。ax² 称为二次项，bx 称为一次项，c 称为常数项。

Examples of quadratic equations: x² – 5x + 6 = 0, 2x² + 3x – 2 = 0, and x² = 9. Notice that the last example has no linear term and no constant term, but it is still quadratic because the highest power of x is 2.

二次方程的例子：x² – 5x + 6 = 0、2x² + 3x – 2 = 0、x² = 9。注意最后一个例子没有一次项和常数项，但它仍然是二次方程，因为 x 的最高次数为 2。

2. Writing Equations in Standard Form | 将方程化为标准形式

Before solving any quadratic equation, you must rearrange it so that one side equals zero. This is called the standard form.

求解任何二次方程之前，必须先整理方程，使一侧等于 0，这称为标准形式。

Example: Solve x² = 7x – 10. First, bring all terms to the left:

例：解 x² = 7x – 10。先把所有项移到左边：

x² – 7x + 10 = 0

Now the coefficients are a = 1, b = -7, c = 10. Notice that the sign of each term is carried with it.

此时系数为 a = 1，b = -7，c = 10。注意每一项的符号要随项一起移动。

Always check that the x² coefficient is positive. If it is negative, multiply the whole equation by -1 for easier factorisation. For example, -x² + 4x – 3 = 0 is better rewritten as x² – 4x + 3 = 0 before solving.

务必检查 x² 的系数是否为正。若为负，可将整个方程乘以 -1，这样更容易分解因式。例如，-x² + 4x – 3 = 0 最好先改写成 x² – 4x + 3 = 0 再求解。

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

Factorisation is the quickest method when the quadratic expression can be written as a product of two linear brackets. It relies on the zero product property: if p × q = 0, then p = 0 or q = 0.

当二次式能写成两个一次括号相乘时，因式分解是最快捷的方法。其依据是零积性质：若 p × q = 0，则 p = 0 或 q = 0。

Example: Solve x² – 7x + 10 = 0. We look for two numbers that multiply to +10 and add to -7. Those numbers are -2 and -5.

例：解 x² – 7x + 10 = 0。我们要找两个数，相乘得 +10，相加得 -7。这两个数是 -2 和 -5。

(x – 2)(x – 5) = 0

Hence x – 2 = 0 or x – 5 = 0, so x =

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）