Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper. Mastering this topic requires a clear understanding of algebraic methods, the discriminant, and graphical interpretation. This revision guide will walk you through every essential skill step by step.

二次方程几乎是每一份 IGCSE 数学试卷中都会出现的考点。掌握这一主题，需要清楚理解代数方法、判别式以及图像意义。本复习指南将一步步带你掌握每一个核心技巧。

1. Introduction to Quadratic Equations | 二次方程简介

A quadratic equation is an equation in which the highest power of the unknown is 2. Its general form is shown below.

二次方程是未知数最高次数为 2 的方程，其一般形式如下。

ax² + bx + c = 0, where a ≠ 0

Here, a is the coefficient of x², b is the coefficient of x, and c is a constant term. Every quadratic equation has up to two solutions, which are called roots.

其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。每个二次方程最多有两个解，称为根。

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

Before solving any quadratic equation, always rearrange it into the standard form ax² + bx + c = 0. This makes it easy to identify coefficients and choose the best method.

在解任何二次方程之前，一定要先将它整理成标准形式 ax² + bx + c = 0。这样可以方便地确定各项系数，并选择最适合的方法。

Equation	Rearranged form	a	b	c
2x² + 3x = 5	2x² + 3x − 5 = 0	2	3	−5
x² = 4 − 2x	x² + 2x − 4 = 0	1	2	−4

Notice that the sign of each term is included. A negative constant is written with the minus sign.

注意每一项的正负号都要保留。当常数项为负数时，要写上负号。

3. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic has simple integer factors. The key principle is the zero product property: if the product of two factors is 0, then at least one factor must be 0.

当二次式具有简单的整数因式时，因式分解是最快捷的方法。其核心原理是零乘积性质：如果两个因式的乘积为 0，那么至少有一个因式必须为 0。

Example: solve x² − 5x + 6 = 0.

例：解方程 x² − 5x + 6 = 0。

(x − 2)(x − 3) = 0

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

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

Always expand your answer to check that the factors produce the original equation.

求解后一定要展开因式来检查是否与原方程一致。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method is especially useful when the quadratic cannot be factorised easily.

配方法将二次式改写为 (x + p)² + q 的形式。当二次式不易因式分解时，这种方法特别有用。

Example: solve x² + 6x + 2 = 0.

例：解方程 x² + 6x + 2 = 0。

First, halve the coefficient of x: 6 ÷ 2 = 3. Then write:

首先，将 x 的系数取半：6 ÷ 2 = 3。然后写出：

(x + 3)² − 9 + 2 = 0

(x + 3)² = 7

Taking square roots gives x + 3 = ±√7, so x = −3 ± √7.

两边开平方得 x + 3 = ±√7，因此 x = −3 ± √7。

Remember that the square root of a positive number has two values: one positive and one negative.

记住，一个正数的平方根有两个值：一个正，一个负。

5. The Quadratic Formula | 求根公式

The quadratic formula works for every quadratic equation, including those with irrational or no real roots. It is derived from completing the square.

求根公式适用于所有二次方程，包括具有无理根或无实根的方程。它由配方法推导而来。

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3.

例如，解方程 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。

Substitute carefully:

代入时要小心：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (2 ± √10) / 2

Always simplify the square root and the fraction completely in your final answer.

最终答案中一定要化简平方根和分数。

6. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us the number and type of roots without solving the equation.

求根公式中的 b² − 4ac 称为判别式。它能在不解方程的情况下告诉我们根的数量和类型。

If b² − 4ac > 0, there are two distinct real roots.

若 b² − 4ac > 0，方程有两个不同的实数根。
If b² − 4ac = 0, there is exactly one repeated real root.

若 b² − 4ac = 0，方程有一个相等的实数根（重根）。
If b² − 4ac < 0, there are no real roots.

若 b² − 4ac < 0，方程没有实数根。

This is a favourite exam topic, so practise identifying the discriminant quickly.

这是考试中非常常见的考点，所以要多练习快速计算判别式。

7. Roots and Coefficients | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots relate directly to the coefficients.

对于方程 ax² + bx + c = 0，若其根为 α 和 β，则根的和与积与系数有直接关系。

Sum of roots: α + β = −b/a

Product of roots: αβ = c/a

These relationships let you find the roots when one root is known, or form a new quadratic with given roots.

这些关系可以帮助你在已知一个根时求出另一个根，或者根据给定的根构造一个新的二次方程。

8. Solving Word Problems | 应用题

Quadratic equations often model areas, projectile motion, and number problems. The first step is to translate the words into an equation.

二次方程常用来建立面积、抛体运动以及数字问题的数学模型。第一步是将文字转化为方程。

Example: A rectangle has length 3 cm longer than its width. Its area is 28 cm². Find the width.

例：一个矩形的长比宽多 3 cm，面积为 28 cm²。求宽。

Let width = x. Then length = x + 3, so x(x + 3) = 28.

设宽为 x，则长为 x + 3，所以 x(x + 3) = 28。

x² + 3x − 28 = 0 → (x + 7)(x − 4) = 0 → x = 4

We reject x = −7 because length cannot be negative. Always check whether every algebraic solution makes sense in the context.

我们舍去 x = −7，因为长度不能为负。一定要检查每个代数解是否符合实际意义。

9. Graphical Interpretation | 图像意义

The graph of y = ax² + bx + c is a parabola. The roots of the equation are the x-coordinates of the points where the parabola crosses the x-axis.

二次函数 y = ax² + bx + c 的图像是抛物线。方程 ax² + bx + c = 0 的根就是抛物线与 x 轴交点的 x 坐标。

When a > 0, the parabola opens upward and has a minimum point.

当 a > 0 时，抛物线开口向上，有最小值点。
When a < 0, the parabola opens downward and has a maximum point.

当 a < 0 时，抛物线开口向下，有最大值点。
When the discriminant is negative, the parabola does not intersect the x-axis.

当判别式为负时，抛物线与 x 轴不相交。

Sketch the graph whenever possible to check that your roots are reasonable.

只要可能就画出草图，以检查你的根是否合理。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students lose marks on quadratic equations because of small but avoidable errors.

许多学生在二次方程上失分，是因为一些细小但可以避免的错误。

Do not forget to rearrange the equation into standard form before applying the formula.

在使用求根公式前，不要忘记将方程整理成标准形式。
When taking square roots, include both the positive and negative values.

开平方时，要同时写出正负两个值。
Check your substitution in the quadratic formula: misplaced brackets cause mistakes.

检查你在求根公式中的代入：括号位置错误会导致计算失误。
Factorising is quick, but always expand to verify your factors.

因式分解虽然快速，但一定要展开验证你的因式。
In word problems, reject any root that does not fit the physical situation.

在应用题中，要舍去不符合实际情境的根。

11. Practice Questions | 练习

Test yourself on these typical IGCSE questions.

请完成下面这些典型的 IGCSE 练习题。

Solve x² − 3x − 10 = 0 by factorisation.

用因式分解法解 x² − 3x − 10 = 0。
Solve 2x² + 5x − 1 = 0 using the quadratic formula, giving your answers correct to 2 decimal places.

用求根公式解 2x² + 5x − 1 = 0，答案精确到两位小数。
Find the values of k such that the equation kx² + 4x + 1 = 0 has two distinct real roots.

求使得方程 kx² + 4x + 1 = 0 有两个不同实数根时的 k 值范围。

Write your working out step by step, then check your answers using substitution.

写出逐步计算过程，然后用代入法检查你的答案。

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Solving Quadratic Equations | 解二次方程