Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear frequently in IGCSE Mathematics. In this revision guide, we will explore the main methods of solving them, understand the discriminant, and look at common exam-style questions.

一元二次方程在 IGCSE 数学中非常常见。在本复习指南中，我们将探讨解这类方程的主要方法，理解判别式，并分析常见考试题型。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0, and a, b, c are real numbers. The highest power of x is 2, which gives the equation its name.

一元二次方程是形如 ax² + bx + c = 0 的方程，其中 a ≠ 0，且 a、b、c 为实数。x 的最高次数是 2，这也是方程名称的由来。

The standard form is ax² + bx + c = 0.
标准形式为 ax² + bx + c = 0。
If a = 0, the equation becomes linear, not quadratic.
如果 a = 0，方程就变成一次方程，而不是二次方程。
Solutions are also called roots.
解也称为根。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic can be written as a product of two linear factors. To solve, set each factor equal to zero.

当一个二次式能写成两个一次因式的乘积时，因式分解通常是最快的方法。求解时，令每个因式等于零。

If (x – p)(x – q) = 0, then x = p or x = q.

例如，解方程 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0，所以 x = 2 或 x = 3。

First, rearrange the equation into standard form.
首先，将方程化为标准形式。
Then factorise the left-hand side.
然后对左边进行因式分解。
Finally, apply the zero product property.
最后，使用零乘积性质。

3. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, use the quadratic formula. This formula works for all quadratic equations.

当因式分解困难或无法进行时，可使用求根公式。该公式适用于所有一元二次方程。

x = (-b ± √(b² – 4ac)) / (2a)

To use it, identify a, b, and c from ax² + bx + c = 0, then substitute into the formula.

使用时，先从 ax² + bx + c = 0 中找出 a、b、c，然后代入公式。

Be careful with negative signs when substituting.
代入时注意负号。
The symbol ± means two possible values: one with plus, one with minus.
符号 ± 表示两个可能的值：一个加，一个减。
No need to simplify the square root if it is not a perfect square.
如果根号下不是完全平方数，则无需化简。

4. Completing the Square | 配方法

Completing the square rewrites x² + bx as (x + b/2)² – (b/2)². This method is useful for solving equations and for finding the turning point of a parabola.

配方法将 x² + bx 改写为 (x + b/2)² – (b/2)²。这种方法既可用于解方程，也可用于求抛物线的顶点。

x² + bx = (x + b/2)² – (b/2)²

For example, solve x² + 6x + 4 = 0. Write (x + 3)² – 9 + 4 = 0, so (x + 3)² = 5. Then x + 3 = ±√5, giving x = -3 ± √5.

例如，解 x² + 6x + 4 = 0。写成 (x + 3)² – 9 + 4 = 0，即 (x + 3)² = 5。于是 x + 3 = ±√5，得到 x = -3 ± √5。

Use this method when the coefficient of x² is 1 or can be made 1.
当 x² 的系数为 1 或可化为 1 时使用此方法。
The turning point of y = (x + p)² + q is (-p, q).
y = (x + p)² + q 的顶点为 (-p, q)。

5. The Discriminant | 判别式

The discriminant is the expression b² – 4ac inside the square root of the quadratic formula. It tells us how many real roots an equation has.

判别式是求根公式中根号内的表达式 b² – 4ac。它告诉我们方程有多少个实数根。

Discriminant b² – 4ac	Number of real roots
Positive (> 0)	Two distinct real roots
Zero (= 0)	One repeated real root
Negative (< 0)	No real roots

In the first two rows above, “real” means the roots are ordinary numbers. A negative discriminant indicates the roots are complex, which are not required in the Edexcel IGCSE core syllabus.

上表前两行中的“实数”表示根是普通数。负的判别式表明根为复数，这在 Edexcel IGCSE 核心考纲中不作要求。

6. Solving by Graphing | 图像法

You can solve a quadratic equation by drawing its graph and reading the x-intercepts. The x-coordinates of the points where the curve crosses the x-axis are the roots.

你可以通过绘制二次函数的图像并读取它与 x 轴的交点来解方程。曲线与 x 轴交点的横坐标就是方程的根。

The graph y = ax² + bx + c is a parabola.
y = ax² + bx + c 的图像是抛物线。
If the parabola touches the x-axis at one point, the equation has one repeated root.
如果抛物线与 x 轴相切于一点，则方程有一个重根。
If it does not touch the x-axis, there are no real roots.
如果它不与 x 轴相交，则没有实数根。
Graphical solutions are often approximate; use them for checking.
图像法得到的解通常为近似值，可用于检查。

7. Solving Word Problems | 应用题

Many IGCSE problems involve forming a quadratic equation from a real-life situation. Read the question carefully, define a variable, and translate the conditions into an equation.

许多 IGCSE 题目要求从实际情境中建立二次方程。仔细阅读题目，设一个变量，并将条件转化为方程。

Example: The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Find the width.

例如：一个长方形的长比宽多 3 cm，面积为 40 cm²。求宽。

Let width = w, then length = w + 3. Area gives w(w + 3) = 40, so w² + 3w – 40 = 0. Factorise to (w + 8)(w – 5) = 0, hence w = 5 (reject w = -8).

设宽为 w，则长为 w + 3。面积得 w(w + 3) = 40，即 w² + 3w – 40 = 0。因式分解为 (w + 8)(w – 5) = 0，因此 w = 5（舍去 w = -8）。

Always check that your answer makes sense in context.
务必检查答案是否符合实际背景。
Discard negative lengths or times unless they are meaningful.
除非有实际意义，否则应舍去负的长度或时间。

8. Common Mistakes | 常见错误

Students often lose marks in this topic through small but avoidable errors. Here are the most common ones.

学生在这一主题中常因一些细小但可避免的错误而丢分。以下是最常见的几种。

Forgetting to set the equation to zero before factorising.
在因式分解之前忘记将方程化为零。
Misidentifying a, b, and c when using the quadratic formula.
使用求根公式时弄错 a、b、c。
Incorrectly expanding (x + p)² as x² + p².
错误地将 (x + p)² 展开为 x² + p²。
Forgetting the ± sign when taking square roots.
开平方时忘记 ± 符号。
Not showing all working when using a calculator.
使用计算器时不展示完整过程。

9. Exam-Style Practice | 考试型练习

Try these questions on your own, then check your answers. They cover the main methods from this guide.

请先独立尝试这些题目，然后核对答案。它们涵盖了本指南中的主要方法。

Solve x² – 7x + 12 = 0.
解方程 x² – 7x + 12 = 0。
Solve 2x² + 5x – 3 = 0 using the quadratic formula.
用求根公式解 2x² + 5x – 3 = 0。
Find the roots of x² – 4x – 1 = 0 by completing the square.
通过配方法求 x² – 4x – 1 = 0 的根。

Answers: 1) x = 3 or x = 4 2) x = 0.5 or x = -3 3) x = 2 ± √5

10. Summary | 总结

Choose the most efficient method for each quadratic. Factorise whenever possible, otherwise use the quadratic formula. Remember that the discriminant reveals the number of real roots.

对于每个二次方程，选择最有效的方法。能因式分解就因式分解，否则使用求根公式。记住判别式揭示了实数根的个数。

Factorisation is fastest for simple integer roots.
对于简单的整数根，因式分解最快。
The quadratic formula works for all quadratics.
求根公式适用于所有二次方程。
Completing the square is useful for vertex form and exact surd answers.
配方法适用于顶点形式和精确根式答案。
Always check answers by substitution into the original equation.
总是通过代入原方程来检验答案。

Practice regularly with past papers to build speed and confidence. Good luck with your revision!

定期用历年真题练习，以提高速度和信心。祝你复习顺利！

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Solving Quadratic Equations | 解一元二次方程