Solving Quadratic Equations | 一元二次方程的解法

📚 Solving Quadratic Equations | 一元二次方程的解法

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, coordinate geometry, functions, and many examination word problems. A solid understanding of how to solve quadratic equations will help you score valuable marks and prepare you for AS & A Level mathematics.

一元二次方程是 IGCSE 数学大纲中最重要的话题之一。它不仅出现在代数、坐标几何和函数中，也是许多考试应用题的核心。牢固掌握一元二次方程的解法，能帮助你在考试中拿到宝贵分数，并为 AS & A Level 数学打下坚实基础。

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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. It has no term with a higher degree such as x³ or x⁴.

一元二次方程是指未知数的最高次数为 2 的方程，方程中不会出现 x³ 或 x⁴ 等更高次项。

For example, x² − 4x + 3 = 0 and 2x² + 5x − 1 = 0 are quadratic equations, but x³ + x² = 1 is not.

例如，x² − 4x + 3 = 0 和 2x² + 5x − 1 = 0 都是一元二次方程，而 x³ + x² = 1 不是。

Some quadratic equations can be disguised. An equation such as x⁴ − 5x² + 4 = 0 can be turned into a quadratic by substituting u = x².

有些方程看起来不是二次方程，但可以转化成二次方程来解。例如 x⁴ − 5x² + 4 = 0 可以通过令 u = x² 化为关于 u 的二次方程。

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2. The Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

Before solving, a quadratic equation should be rearranged into standard form:

在求解之前，应先把一元二次方程整理成标准形式：

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants. The condition a ≠ 0 is essential: if a = 0, the equation becomes linear.

其中 a、b、c 为常数。条件 a ≠ 0 至关重要：如果 a = 0，方程就变成了一次方程。

For example, to solve 5x = 6 − 2x², you first rearrange to 2x² + 5x − 6 = 0. Always collect all terms on one side of the equation.

例如，解 5x = 6 − 2x² 时，要先整理成 2x² + 5x − 6 = 0。记住：一定要把所有项移到等号同一边。

It is also good practice to arrange the terms in descending powers: the x² term first, then the x term, then the constant.

同时建议按降幂排列：先写 x² 项，再写 x 项，最后写常数项。

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3. Solving by Factorisation | 因式分解法

Factorisation is usually the fastest method when the quadratic has simple integer factors. The idea is to write the expression as a product of two brackets.

当二次式具有简单的整数因子时，因式分解法通常是最快的解法。核心思路是把表达式写成两个括号相乘的形式。

Follow these steps:

解题步骤如下：

• Rearrange the equation into the form ax² + bx + c = 0.

  将方程整理为 ax² + bx + c = 0 的标准形式。

• Write the quadratic expression as (px + m)(qx + n).

  把二次表达式写成 (px + m)(qx + n) 的形式。

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

  利用零积性质：若 AB = 0，则 A = 0 或 B = 0。

• Solve the two resulting linear equations.

  分别解出两个一次方程。

Example: Solve x² − 5x + 6 = 0.

例：解方程 x² − 5x + 6 = 0。

We look for two numbers that multiply to 6 and add to −5. The numbers are −2 and −3, so the factorised form is (x − 2)(x − 3) = 0.

寻找两个数：它们的乘积为 6，和为 −5。这两个数是 −2 和 −3，因此分解结果为 (x − 2)(x − 3) = 0。

Therefore x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

所以 x − 2 = 0 或 x − 3 = 0，解得 x = 2 或 x = 3。

For quadratics where a ≠ 1, you may use the method of factors of ac. For example, in 2x² + 7x + 3 = 0, two numbers that multiply to 2 × 3 = 6 and add to 7 are 1 and 6, so we split the middle term: 2x² + x + 6x + 3 = 0, then factor by grouping.

对于 a ≠ 1 的情况，可以使用 ac 值分组分解法。例如在 2x² + 7x + 3 = 0 中，找两个数乘积为 2 × 3 = 6，和为 7，即 1 和 6；拆中项得 2x² + x + 6x + 3 = 0，再分组提取公因式。

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4. Solving by the Quadratic Formula | 公式法求解

If a quadratic cannot be factorised easily, the quadratic formula always works. For the equation ax² + bx + c = 0:

如果二次式不容易分解，可以用二次公式，它适用于一切一元二次方程。对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / (2a)

Example: Solve x² + 3x − 4 = 0 using the formula.

例：用公式法解 x² + 3x − 4 = 0。

Here a = 1, b = 3 and c = −4. Substitute into the formula:

这里 a = 1，b = 3，c = −4

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