Quadratic Equations: Solving, Graphing, and Applications | 二次方程：求解、图像与应用

📚 Quadratic Equations: Solving, Graphing, and Applications | 二次方程：求解、图像与应用

Quadratic equations are a central part of the IGCSE Mathematics syllabus. This article covers the standard form, three solving methods, the discriminant, graph interpretation, and common applications. You will also find worked examples and exam tips.

二次方程是 IGCSE 数学大纲的核心内容。本文涵盖标准形式、三种求解方法、判别式、图像解读以及常见应用。你还会看到例题和应试技巧。

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

In IGCSE Mathematics, a quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable x is 2.

在 IGCSE 数学中，二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数是 2。

If a were equal to 0, the x² term would disappear, leaving a linear equation bx + c = 0. That is why the condition a ≠ 0 matters.

如果 a 等于 0，x² 项就会消失，只剩下一次方程 bx + c = 0。这就是为什么 a ≠ 0 这个条件很重要。

Examples of quadratic equations include x² – 3x + 2 = 0, 4x² – 9 = 0, and 2x² + 5x – 3 = 0.

二次方程的例子包括 x² – 3x + 2 = 0、4x² – 9 = 0 和 2x² + 5x – 3 = 0。

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

The standard form of a quadratic equation is ax² + bx + c = 0. Here a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.

二次方程的标准形式是 ax² + bx + c = 0。其中 a 是二次项系数，b 是一次项系数，c 是常数项。

Before solving, always rearrange the equation so that all terms are on one side and the other side is zero. For example, 3x² = 5x – 2 becomes 3x² – 5x + 2 = 0.

求解前，一定要把方程整理成所有项都在一边、另一边为零的形式。例如，3x² = 5x – 2 应变为 3x² – 5x + 2 = 0。

Identifying a, b and c correctly is very important when using the quadratic formula or discriminant.

正确识别 a、b 和 c 在使用二次公式或判别式时非常重要。

3. Solving by Factorising | 因式分解法

Factorising is often the quickest method when the quadratic has integer roots. You look for two numbers that multiply to give ac and add to give b, then split the middle term or write two brackets directly.

当二次方程有整数根时，因式分解通常是最快的方法。你需要找到两个数，它们的乘积为 ac，和为 b，然后拆分一次项或直接写出两个括号。

Worked example: Solve x² – 7x + 12 = 0.

例题：解 x² – 7x + 12 = 0。

The two numbers are -3 and -4 because (-3) × (-4) = 12 and (-3) + (-4) = -7. Therefore (x – 3)(x – 4) = 0, so x = 3 or x = 4.

这两个数是 -3 和 -4，因为 (-3) × (-4) = 12 且 (-3) + (-4) = -7。因此 (x – 3)(x – 4) = 0，所以 x = 3 或 x = 4。

Always verify by expanding: (x – 3)(x – 4) = x² – 4x – 3x + 12 = x² – 7x + 12.

一定要通过展开来验证：(x – 3)(x – 4) = x² – 4x – 3x + 12 = x² – 7x + 12。

Another example: Solve 2x² + 5x – 3 = 0. Here ac = -6, and the two numbers are 6 and -1. Split the middle term: 2x² + 6x – x – 3 = 0, then factor by grouping: 2x(x + 3) – 1(x + 3) = 0, so (2x – 1)(x + 3) = 0. Thus x = 1/2 or x = -3.

另一个例子：解 2x² + 5x – 3 = 0。这里 ac = -6，两个数是 6 和 -1。拆分一次项：2x² + 6x – x – 3 = 0，然后分组分解：2x(x + 3) – 1(x + 3) = 0，所以 (2x – 1)(x + 3) = 0。因此 x = 1/2 或 x = -3。

4. Completing the Square | 配方法

Completing the square changes the expression x² + bx into (x + b/2)² – (b/2)². This technique is required for deriving the quadratic formula and for sketching graphs.

配方法将表达式 x² + bx 变为 (x + b/2)² – (b/2)²。这个技巧是推导二次公式和绘制图像所需要的。

Example: Solve

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