📚 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, 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
xample, 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。
) = 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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