Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, graphs, geometry and word problems. This revision guide will take you through the standard form, four solving methods, the discriminant, and common errors, so that you can answer any quadratic question with confidence.

一元二次方程是 IGCSE 数学考纲中最重要的内容之一，它出现在代数、图像、几何和应用题中。本复习指南将带你掌握标准形式、四种解法、判别式以及常见错误，使你能够自信地解答任何一元二次方程问题。

1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is an equation in which the highest power of the unknown variable is 2. For example, x² − 4x + 3 = 0 is a quadratic equation, while x³ − 1 = 0 is not.

一元二次方程是未知数的最高次数为 2 的方程。例如，x² − 4x + 3 = 0 是一元二次方程，而 x³ − 1 = 0 不是。

In IGCSE Mathematics, you will meet equations that can be rearranged into this form. The word ‘quadratic’ comes from the Latin word ‘quadratus’, which means square.

在 IGCSE 数学中，你会遇到许多可以化为这种形式的方程。”Quadratic” 一词源自拉丁语 “quadratus”，意为”平方”。

ax² + bx + c = 0, a ≠ 0

The letters a, b and c are constants, and x is the variable. The condition a ≠ 0 is essential: if a = 0, the equation becomes linear, not quadratic.

字母 a、b 和 c 是常数，x 是变量。条件 a ≠ 0 至关重要：若 a = 0，原方程就变成了一次方程，而不是二次方程。

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

Before solving any quadratic equation, you must write it in standard form: ax² + bx + c = 0. This means every term is on one side of the equals sign, and the other side is zero.

在求解任何一元二次方程之前，必须先将其写成标准形式 ax² + bx + c = 0，即所有项都在等号的一侧，另一侧为零。

Remember that the coefficient includes the sign in front of the term. For example, in 2x² + 3x − 5 = 0, the value of c is −5, not 5.

请记住：系数包含该项前面的符号。例如，在 2x² + 3x − 5 = 0 中，常数 c = −5，而不是 5。

Some equations require expansion before rearranging. For instance, (x + 1)² = 4 must first be expanded to x² + 2x + 1 = 4, then rearranged to x² + 2x − 3 = 0.

有些方程需要先展开再整理。例如，(x + 1)² = 4 必须先展开为 x² + 2x + 1 = 4，再整理为 x² + 2x − 3 = 0。

3. Method 1: Solving by Factorisation | 方法一：因式分解法

Factorisation is the fastest method when the quadratic expression can be written as the product of two linear factors.

当二次表达式可以写成两个一次因式的乘积时，因式分解法是最快的解法。

Follow these steps:

请按以下步骤操作：

• Rearrange the equation so that one side is zero.

  将方程整理为一边为零。

• Factorise the quadratic expression.

  对二次表达式进行因式分解。

• Use the zero product property: if A × B = 0, then A = 0 or B = 0.

  运用零乘积性质：若 A × B = 0，则 A = 0 或 B = 0。

• Solve the two linear equations.

  分别解这两个一次方程。

Worked example:

示例：

x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3

Check the expansion: (x − 2)(x − 3) = x² − 3x − 2x + 6 = x² − 5x + 6. The factorisation is correct.

验证展开式：(x − 2)(x − 3) = x² − 3x − 2x + 6 = x² − 5x + 6。因式分解正确。

If the coefficient a is not 1, look for a common factor first. For example, 2x² + 8x + 6 = 0 can be divided throughout by 2 to give x² + 4x + 3 = 0.

如果二次项系数 a 不为 1，先寻找公因式。例如，2x² +

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