Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in many exam problems, from simple factorisation to more complex word problems. In this revision guide, we will explore the standard methods of solving quadratic equations, including factorising, completing the square, using the quadratic formula, and interpreting graphs. We will also discuss the discriminant and common mistakes to help you secure full marks.

二次方程是 IGCSE 数学大纲中最重要的主题之一。从简单的因式分解到复杂的应用题，它们出现在许多考试题目中。在本复习指南中，我们将探讨解二次方程的标准方法，包括因式分解、配方法、使用求根公式以及解释图像。我们还将讨论判别式和常见错误，帮助你拿到满分。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants, and a is not equal to 0. The highest power of the variable x is 2, which is why it is called ‘quadratic’ (from the Latin word quadratus meaning square).

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a 不等于 0。变量的最高次数是 2，因此被称为“二次”（来自拉丁语 quadratus，意思是平方）。

In the Edexcel IGCSE syllabus, you are expected to solve quadratic equations using a variety of methods, and to choose the most efficient method in each question.

在 Edexcel IGCSE 大纲中，你需要用多种方法解二次方程，并在每题中选择最有效的方法。

2. Standard Form and Key Terms | 标准形式与关键术语

Before solving, always rearrange the equation into the standard form ax² + bx + c = 0. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.

在求解之前，请先将方程整理成标准形式 ax² + bx + c = 0。其中 ax² 称为二次项，bx 称为一次项，c 称为常数项。

The solutions of a quadratic equation are the values of x that make the equation true. They are also called roots or zeros of the equation.

二次方程的解是使方程成立的 x 的值。它们也被称为方程的根或零点。

3. Solving by Factorising | 因式分解法

Factorising is often the fastest method when the quadratic can be written as a product of two linear expressions. For example, x² + 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0.

当二次式可以写成两个一次表达式的乘积时，因式分解通常是最快的方法。例如，x² + 5x + 6 = 0 可以分解为 (x + 2)(x + 3) = 0。

Using the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. Hence we set each factor equal to zero and solve.

利用零乘积性质：如果两个因式的乘积为零，那么至少有一个因式必须为零。因此，我们令每个因式等于零并求解。

(x + 2)(x + 3) = 0 ⇒ x + 2 = 0 or x + 3 = 0 ⇒ x = −2 or x = −3

When the coefficient of x² is not 1, for example 2x² + 7x + 3 = 0, you need to find two numbers that multiply to give a × c and add to give b. Here, the numbers are 6 and 1, so the factorised form is (2x + 1)(x + 3) = 0.

当 x² 的系数不为 1 时，例如 2x² + 7x + 3 = 0，你需要找到两个数，它们相乘等于 a × c，相加等于 b。这里，这两个数是 6 和 1，因此分解形式是 (2x + 1)(x + 3) = 0。

4. Solving by Completing the Square | 配方法

Completing the square is a method that works for any quadratic equation, and it also helps you find the turning point of a parabola. The idea is to rewrite ax² + bx + c in the form a(x + p)² + q, where p and q are constants.

配方法是一种适用于任何二次方程的方法，同时也能帮助你找到抛物线的顶点。其思想是将 ax² + bx + c 改写成 a(x + p)² + q 的形式，其中 p、q 是常数。

For the equation x² + 6x + 5 = 0, we first consider the coefficient of x, which is 6. Half of 6 is 3, so we write (x + 3)² − 9 + 5 = 0, which simplifies to (x + 3)² − 4 = 0.

对于方程 x² + 6x + 5 = 0，我们先看 x 的系数 6。6 的一半是 3，因此我们写成 (x + 3)² − 9 + 5 = 0，化简为 (x + 3)² − 4 = 0。

(x + 3)² = 4 ⇒ x + 3 = ±2 ⇒ x = −1 or x = −5

If the coefficient of x² is not 1, factor it out first before completing the square.

如果 x² 的系数不是 1，请先将其提取出来，再进行配方。

5. Solving by the Quadratic Formula | 公式法

The quadratic formula is a universal method that works for every quadratic equation, even when factorising is difficult or impossible. The formula is given in the Edexcel IGCSE formula sheet, but you must know how to use it correctly.

求根公式是一种通用方法，适用于所有二次方程，即使因式分解困难或不可能也能使用。该公式在 Edexcel IGCSE 公式表中给出，但你必须知道如何正确使用它。

x = (−b ± √(b² − 4ac)) / (2a)

To use the formula, identify the values of a, b and c from the equation in standard form, then substitute them into the formula and simplify.

使用公式时，先从标准形式的方程中识别出 a、b、c 的值，然后代入公式并化简。

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

例如，解 2x² − 3x − 2 = 0。这里 a = 2，b = −3，c = −2。

x = (3 ± √((−3)² − 4 × 2 × (−2))) / (2 × 2) = (3 ± √25) / 4 = (3 ± 5) / 4

x = 2 or x = −0.5

Always check whether your answers make sense by substituting them back into the original equation.

始终将答案代回原方程检查是否合理。

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 completely.

求根公式中的表达式 b² − 4ac 称为判别式。它告诉我们根的数量和类型，而不需要完全解方程。

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

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

如果 b² − 4ac = 0，方程只有一个实数根（重根）。
If b² − 4ac < 0, there are no real roots (two complex roots).

如果 b² − 4ac < 0，方程没有实数根（有两个复数根）。

The discriminant is also useful when solving problems about tangents to curves or intersection of lines and parabolas.

判别式在解决关于曲线切线或直线与抛物线交点的问题时也很有用。

7. Solving Quadratic Equations by Graphs | 图像法

You can solve a quadratic equation by drawing the graph of y = ax² + bx + c and reading the x-coordinates where the curve crosses the x-axis. These points are the roots of the equation ax² + bx + c = 0.

你可以通过绘制 y = ax² + bx + c 的图像，并读取曲线与 x 轴交点的 x 坐标来解二次方程。这些交点就是方程 ax² + bx + c = 0 的根。

If the curve touches the x-axis at one point, the equation has a repeated root. If the curve does not meet the x-axis at all, the equation has no real roots.

如果曲线与 x 轴只相切于一点，则方程有一个重根。如果曲线完全不与 x 轴相交，则方程没有实数根。

For example, the graph of y = x² − 2x − 3 crosses the x-axis at x = −1 and x = 3, so the equation x² − 2x − 3 = 0 has roots −1 and 3.

例如，y = x² − 2x − 3 的图像在 x = −1 和 x = 3 处与 x 轴相交，因此方程 x² − 2x − 3 = 0 的根为 −1 和 3。

8. Word Problems | 应用题

In IGCSE exams, quadratic equations often appear in word problems. You need to translate the problem into an equation, solve it, and then interpret the answers in the context of the question. Be careful: sometimes one root must be rejected because it does not make sense (for example, a negative length).

在 IGCSE 考试中，二次方程通常出现在应用题中。你需要将实际问题转化为方程，求解，然后在问题情境中解释答案。注意：有时某个根必须舍去，因为它没有意义（例如长度为负数）。

Example: The area of a rectangle is 24 cm², and its length is 2 cm more than its width. Find the width.

例：一个矩形的面积是 24 cm²，其长比宽多 2 cm。求宽。

Let width = x ⇒ length = x + 2 ⇒ x(x + 2) = 24 ⇒ x² + 2x − 24 = 0

(x + 6)(x − 4) = 0 ⇒ x = −6 or x = 4

Since a width cannot be negative, the width is 4 cm.

由于宽度不能为负，所以宽为 4 cm。

9. Common Mistakes and Tips | 常见错误与技巧

One common mistake is forgetting to rearrange the equation to the form ax² + bx + c = 0 before factorising or using the formula. Always move all terms to one side first.

一个常见错误是忘记将方程整理成 ax² + bx + c = 0 的形式，然后再进行因式分解或使用公式。永远先把所有项移到一边。

Another mistake is losing the square root sign when using the quadratic formula. Remember to take the positive and negative square roots, and simplify carefully.

另一个错误是使用求根公式时丢掉根号。记住要取正负平方根，并仔细化简。

Tip: If a quadratic looks easy to factorise, try factorising first. If the numbers are large or the equation has irrational roots, use the quadratic formula or completing the square.

技巧：如果二次式看起来容易分解，先尝试因式分解。如果数字很大或根是无理数，则使用求根公式或配方法。

Always check your solutions by substituting them back into the original equation. This can help you catch arithmetic errors before the examiners do.

始终将解代回原方程验证。这可以帮助你在考官发现之前找出计算错误。

10. Practice Questions | 练习题

Here are a few practice questions to test your understanding. Try to solve them using different methods.

这里有几道练习题来测试你的理解。尝试用不同的方法解答。

1. Solve x² − 7x + 12 = 0 by factorising.

1. 用因式分解法解 x² − 7x + 12 = 0。
2. Solve 3x² + 5x − 2 = 0 using the quadratic formula.

2. 用求根公式解 3x² + 5x − 2 = 0。
3. Find the discriminant of x² + 4x + 5 = 0 and state the number of real roots.

3. 求 x² + 4x + 5 = 0 的判别式，并说明实数根的个数。
4. The product of two consecutive integers is 42. Find the integers.

4. 两个连续整数的乘积是 42。求这两个整数。

Answers: 1. x = 3, 4. 2. x = ⅓ or x = −2. 3. Δ = −4, no real roots. 4. 6 and 7, or −7 and −6.

答案：1. x = 3, 4。2. x = ⅓ 或 x = −2。3. Δ = −4，无实数根。4. 6 和 7，或 −7 和 −6。

11. Summary | 总结

Quadratic equations can be solved by factorising, completing the square, using the quadratic formula, or reading from a graph. The discriminant b² − 4ac tells you how many real roots exist. Always write the equation in standard form first, choose the most efficient method, and check your answers.

二次方程可以通过因式分解、配方法、求根公式或看图来求解。判别式 b² − 4ac 告诉你存在多少个实数根。始终先将方程写成标准形式，选择最有效的方法，并检查你的答案。

With regular practice, solving quadratic equations will become quick and accurate, helping you gain confidence for the IGSCE Mathematics exam.

通过经常练习，解二次方程将变得快速而准确，帮助你在 IGCSE 数学考试中建立信心。

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Solving Quadratic Equations | 解二次方程