Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratic equations are one of the most important topics in Edexcel IGCSE Mathematics. They appear in almost every exam paper, whether as standalone questions or as part of problem-solving tasks. Mastering the different methods of solving quadratic equations is essential for achieving a high grade.

二次方程是 Edexcel IGCSE 数学中最重要的内容之一。几乎每份试卷都会出现二次方程，无论是独立题目还是作为应用题的一部分。掌握解二次方程的不同方法，是取得高分的关键。

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

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, which makes the graph a parabola.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2，因此其图像是抛物线。

For example, 2x² − 5x + 3 = 0 is a quadratic equation. If a = 0, the equation becomes linear, not quadratic.

例如，2x² − 5x + 3 = 0 就是一个二次方程。如果 a = 0，方程就变成一次方程，而不是二次方程。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic expression can be factorised into two linear factors. The principle is: if the product of two expressions is zero, then at least one of the expressions must be zero.

当二次式可以因式分解为两个一次因式时，因式分解法通常是最快的方法。其原理是：如果两个表达式的乘积为零，则至少有一个表达式为零。

Example: Solve x² − 7x + 10 = 0.

例题：解方程 x² − 7x + 10 = 0。

Steps: Write x² − 7x + 10 = (x − 2)(x − 5). Then set each factor to zero: x − 2 = 0 or x − 5 = 0. Thus x = 2 or x = 5.

步骤：将 x² − 7x + 10 写成 (x − 2)(x − 5)。然后令每个因式为零：x − 2 = 0 或 x − 5 = 0。所以 x = 2 或 x = 5。

Always check whether the coefficient of x² is 1 or not. For example, 2x² + 5x − 3 = (2x − 1)(x + 3). Then x = ½ or x = −3.

注意 x² 的系数是否为 1。例如，2x² + 5x − 3 = (2x − 1)(x + 3)，因此 x = ½ 或 x = −3。

3. Solving by Square Roots | 平方根法

If a quadratic equation has no linear term (b = 0), we can solve it by taking square roots. The form is ax² + c = 0.

如果二次方程没有一次项（b = 0），我们可以通过开平方根来求解。其形式为 ax² + c = 0。

Example: Solve 3x² − 27 = 0.

例题：解方程 3x² − 27 = 0。

First, add 27 to both sides: 3x² = 27. Then divide by 3: x² = 9. Taking square roots gives x = ±3.

首先，两边同时加 27：3x² = 27。然后除以 3：x² = 9。开平方根得到 x = ±3。

Remember that both positive and negative roots must be considered. However, if the right-hand side is negative, there are no real solutions.

记住，正根和负根都必须考虑。然而，如果等号右边为负数，则没有实数解。

4. Completing the Square | 配方法

Completing the square is a powerful algebraic technique that rewrites a quadratic expression in the form p(x + q)² + r. It is especially useful when the equation cannot be factorised easily.

配方法是一种强有力的代数技巧，它将二次式重写为 p(x + q)² + r 的形式。当方程不容易因式分解时，配方法尤其有用。

Example: Solve x² + 6x + 1 = 0 by completing the square.

例题：用配方法解方程 x² + 6x + 1 = 0。

Write x² + 6x = (x + 3)² − 9. So the equation becomes (x + 3)² − 9 + 1 = 0, which simplifies to (x + 3)² = 8. Taking square roots: x + 3 = ±√8, so x = −3 ± 2√2.

先写 x² + 6x = (x + 3)² − 9。于是原方程变为 (x + 3)² − 9 + 1 = 0，即 (x + 3)² = 8。开平方根：x + 3 = ±√8，所以 x = −3 ± 2√2。

For a general quadratic ax² + bx + c = 0, first divide by a, then complete the square inside the brackets.

对于一般的一元二次方程 ax² + bx + c = 0，先除以 a，再在括号内配方。

5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation directly. For ax² + bx + c = 0, the solutions are:

求根公式可以直接求解任何二次方程。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) / (2a)

This formula always works, including when the solutions are irrational or complex (though complex solutions are not required in IGCSE).

这个公式始终有效，包括解为无理数的情况（尽管 IGCSE 不要求复数解）。

Example: Solve 2x² − 4x − 3 = 0.

例题：解方程 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substitute into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

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

So the two solutions are approximately 2.58 and −0.58, depending on whether you add or subtract the square root.

因此两个解大约为 2.58 和 −0.58，取决于你加还是减根号。

6. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells us how many real solutions a quadratic equation has.

根号内的表达式 b² − 4ac 称为判别式。它告诉我们二次方程有多少个实数解。

If b² − 4ac > 0: two distinct real solutions.
如果 b² − 4ac > 0：两个不同的实数解。
If b² − 4ac = 0: exactly one real solution (a repeated root).
如果 b² − 4ac = 0：恰好一个实数解（重根）。
If b² − 4ac < 0: no real solutions.
如果 b² − 4ac < 0：没有实数解。

For example, the equation x² + 2x + 5 = 0 has discriminant 4 − 20 = −16, so it has no real solutions.

例如，方程 x² + 2x + 5 = 0 的判别式为 4 − 20 = −16，因此没有实数解。

7. Solving Word Problems | 解应用题

Quadratic equations often arise from geometric or real-life problems. The key is to translate the English text into an algebraic equation, then solve it using one of the methods above.

二次方程经常出现在几何或实际问题中。关键是将文字描述转化为代数方程，然后用上述一种方法求解。

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

例题：一个长方形的长比宽多 3 cm，面积为 40 cm²。求宽。

Let the width be w. Then length = w + 3. The area equation is w(w + 3) = 40, or w² + 3w − 40 = 0. Factorise: (w + 8)(w − 5) = 0, so w = −8 or w = 5. Since a width cannot be negative, the width is 5 cm.

设宽为 w，则长为 w + 3。面积方程为 w(w + 3) = 40，即 w² + 3w − 40 = 0。因式分解得 (w + 8)(w − 5) = 0，所以 w = −8 或 w = 5。因为宽不能为负数，所以宽为 5 cm。

Always reject negative answers if they do not make sense in the context of the problem.

如果负答案在问题情境中不合理，一定要舍去。

8. Sketching Quadratic Graphs | 画二次函数图像

Solving equations and understanding graphs are closely connected. The roots of a quadratic equation are the x-intercepts of the curve y = ax² + bx + c.

解方程与理解图像密切相关。二次方程的根就是曲线 y = ax² + bx + c 与 x 轴交点的横坐标。

To sketch a quadratic graph, find the roots, the y-intercept (which is c), and the turning point. The turning point lies at x = −b/(2a), and its y-coordinate is found by substituting this x value into the equation.

要画二次函数图像，需要找到根、y 轴截距（即 c）和顶点。顶点的横坐标为 x = −b/(2a)，纵坐标将 x 值代入方程即可得到。

If a > 0, the parabola opens upwards. If a < 0, it opens downwards.

如果 a > 0，抛物线开口向上。如果 a < 0，抛物线开口向下。

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

Students often lose marks by making small errors. Be careful not to forget that a quadratic equation can have two solutions. Do not cancel terms containing x without checking whether x = 0 is a solution. Always check your answers by substituting back into the original equation.

学生常因小错误失分。注意别忘了二次方程可能有两个解。不要在没有检查 x = 0 是否为解的情况下约去含 x 的项。务必通过代入原方程来检查答案。

In the Edexcel IGCSE exam, calculators are allowed in Paper 2 but not in Paper 1. Therefore, you need to know how to solve quadratics both by factorisation and by using the formula manually. Practice completing the square without a calculator as well.

在 Edexcel IGCSE 考试中，Paper 2 允许使用计算器，而 Paper 1 不允许。因此，你需要既掌握因式分解法，又能手算使用求根公式。同时也要练习不用计算器进行配方。

10. Summary and Revision Strategy | 总结与复习策略

To summarise, there are four main methods to solve quadratic equations:

总结一下，解二次方程有四种主要方法：

Method	Best Used When	方法	适用情况
Factorisation	The quadratic factorises easily	因式分解法	二次式容易分解
Square Roots	No x-term (b = 0)	平方根法	没有一次项（b = 0）
Completing the Square	When you need the turning point	配方法	需要求顶点坐标时
Quadratic Formula	Always works, even for non-factorisable	求根公式	任何情况都可用，尤其不能因式分解时

Revise each method by practising past exam questions. Become confident in converting word problems into equations. Remember to use the discriminant to quickly check the nature of roots. Good luck with your revision!

通过练习往年真题来复习每种方法。要熟练地把文字题转化为方程。记得用判别式快速判断根的情况。祝你复习顺利！

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Solving Quadratic Equations | 二次方程求解