Quadratic Equations & Their Solutions | IGCSE数学：二次方程及其解法

📚 Quadratic Equations & Their Solutions | IGCSE数学：二次方程及其解法

Quadratic equations appear frequently in IGCSE Mathematics. In this article, you will learn how to recognise, solve and apply quadratic equations using factorisation, the quadratic formula and completing the square, as well as how to interpret their graphs.

二次方程在IGCSE数学中经常出现。在本文中，你将学习如何识别、求解和应用二次方程，包括因式分解法、求根公式法、配方法，以及如何解读它们的图像。

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

A quadratic equation is an equation that can be written in the standard form:

二次方程是可以写成标准形式的方程：

ax² + bx + c = 0, where a ≠ 0

Here, x is the variable, while a, b and c are constants. The highest power of x is 2, which is why it is called “quadratic”.

其中 x 是变量，a、b、c 为常数。x 的最高次数是 2，因此称为“二次”。

If a = 0, the equation becomes linear, not quadratic.
如果 a = 0，方程变为一次方程，而不是二次方程。
The equation may appear in expanded form, factorised form or with fractions; you may need to rearrange it first.
方程可能以展开式、因式分解式或含分数的形式出现；你可能需要先重新整理。

2. Solving by Factorisation | 因式分解法求解

Factorisation is the fastest method when the quadratic has simple integer roots. It relies on finding two numbers that multiply to give c and add to give b (when a = 1).

因式分解法在二次方程具有简单整数根时最快。它依赖于找到两个数，它们的乘积等于 c，和等于 b（当 a = 1 时）。

If x² + bx + c = (x + p)(x + q) = 0, then x = −p or x = −q

Example: Solve x² − 5x + 6 = 0. We need two numbers whose product is 6 and sum is 5. They are 2 and 3, so (x − 2)(x − 3) = 0. Hence x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。我们需要两个数，乘积为 6，和为 5。这两个数是 2 和 3，因此 (x − 2)(x − 3) = 0。所以 x = 2 或 x = 3。

3. Factorising When a ≠ 1 | 当 a ≠ 1 时的因式分解

When the coefficient of x² is not 1, you must consider factors of a and c together. A common method is the “ac method”: find two numbers that multiply to ac and add to b, then split the middle term.

当 x² 的系数不是 1 时，你必须同时考虑 a 和 c 的因数。常用方法是“ac 法”：找到两个数，它们相乘等于 ac，相加等于 b，然后拆开中间项。

For 2x² + 7x + 3 = 0, ac = 6, b = 7 → 1 and 6

Rewrite as 2x² + 1x + 6x + 3 = 0, then factor by grouping: x(2x + 1) + 3(2x + 1) = 0, so (2x + 1)(x + 3) = 0. Therefore x = −½ or x = −3.

改写为 2x² + 1x + 6x + 3 = 0，然后分组因式分解：x(2x + 1) + 3(2x + 1) = 0，所以 (2x + 1)(x + 3) = 0。因此 x = −½ 或 x = −3。

4. The Quadratic Formula | 求根公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，解由求根公式给出：

x = (−b ± √(b² − 4ac)) / (2a)

This formula works for all quadratics, including those that cannot be factorised easily. You are usually allowed to use it in the exam if you substitute values correctly.

这个公式适用于所有二次方程，包括那些不容易因式分解的。考试中只要代入正确，通常可以使用。

Example: Solve 2x² − 4x − 3 = 0 using the formula.
例：用公式解 2x² − 4x − 3 = 0。
Here a = 2, b = −4, c = −3.
这里 a = 2，b = −4，c = −3。
Substitute: x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± (√10)/2
代入：x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± (√10)/2

5. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells you how many real roots the equation has:

根号内的表达式 b² − 4ac 称为判别式。它告诉你方程有多少个实数根：

Discriminant Δ = b² − 4ac	Nature of roots	根的个数
Δ > 0	Two distinct real roots	两个不同的实数根
Δ = 0	One repeated real root	一个重根（两个相等的实数根）
Δ < 0	No real roots (complex roots)	无实数根（复数根）

Example: For x² + 2x + 5 = 0, Δ = 4 − 20 = −16 < 0, so there are no real solutions. The graph never crosses the x-axis.

例：对 x² + 2x + 5 = 0，Δ = 4 − 20 = −16 < 0，因此没有实数解。图像永不相交于 x 轴。

6. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c as a(x + h)² + k. This is useful for finding the turning point and for solving equations.

配方法将 ax² + bx + c 改写为 a(x + h)² + k。这对于求顶点和解方程都很有用。

x² + bx + c = (x + b/2)² − (b/2)² + c

Example: Solve x² + 6x + 2 = 0 by completing the square.

例：通过配方法解 x² + 6x + 2 = 0。

(x + 3)² − 9 + 2 = 0 → (x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7

7. Solving Quadratic Equations by Graphs | 用图像解二次方程

To solve a quadratic equation graphically, you can plot y = ax² + bx + c and read the x-coordinates where the curve crosses the x-axis. Those x-intercepts are the real roots.

要用图像解二次方程，可以绘制 y = ax² + bx + c 并读取曲线与 x 轴交点的 x 坐标。这些 x 截距就是实数根。

If the curve touches the x-axis at one point, there is one repeated root.
如果曲线与 x 轴仅相切于一点，则有一个重根。
If the curve does not touch the x-axis, the equation has no real roots.
如果曲线不与 x 轴相交，则方程没有实数根。
You can also solve equations like x² = 2x + 3 by plotting y = x² and y = 2x + 3 and finding the intersection points.
你也可以通过绘制 y = x² 和 y = 2x + 3 并找出交点来解像 x² = 2x + 3 这样的方程。

8. The Graph of y = ax² + bx + c | 二次函数的图像

The graph of a quadratic function is a parabola. The sign of a determines its shape:

二次函数的图像是一条抛物线。a 的符号决定其开口方向：

If a > 0: the parabola opens upwards (U-shaped) and has a minimum point.

如果 a > 0：抛物线开口向上（U 形），有最小值点。

If a < 0: the parabola opens downwards (n-shaped) and has a maximum point.

如果 a < 0：抛物线开口向下（n 形），有最大值点。

The turning point lies on the axis of symmetry, which has equation x = −b/(2a). For y = x² − 4x + 3, the axis is x = 2 and the minimum value is y = −1.

顶点位于对称轴上，对称轴方程为 x = −b/(2a)。对于 y = x² − 4x + 3，对称轴为 x = 2，最小值为 y = −1。

9. Word Problems Involving Quadratics | 涉及二次方程的应用题

Many IGCSE problems ask you to form a quadratic equation from a real-world situation. Read the problem carefully, define a variable, then translate the conditions into an equation.

许多IGCSE题目要求你从实际情况中建立二次方程。仔细阅读题目，定义一个变量，然后将条件转化为方程。

Example: A rectangle is 3 cm longer than it is wide, and its area is 40 cm². Let the width be x cm. Then length = x + 3, so x(x + 3) = 40 → x² + 3x − 40 = 0.

例：一个矩形比宽长 3 cm，面积为 40 cm²。设宽为 x cm，则长为 x + 3，因此 x(x + 3) = 40 → x² + 3x − 40 = 0。

Factorise: (x + 8)(x − 5) = 0 → x = −8 or x = 5. Since width cannot be negative, the width is 5 cm and the length is 8 cm.

因式分解得 (x + 8)(x − 5) = 0 → x = −8 或 x = 5。由于宽不能为负，所以宽为 5 cm，长为 8 cm。

10. Rearranging Into Standard Form | 化为标准形式

Not every quadratic equation is given in the form ax² + bx + c = 0. You may need to expand brackets, collect like terms, or remove denominators.

并非每个二次方程都以 ax² + bx + c = 0 的形式给出。你可能需要去括号、合并同类项或去分母。

Example: Solve (x + 1)² = 3x + 7.

例：解 (x + 1)² = 3x + 7。

Expand: x² + 2x + 1 = 3x + 7. Then x² − x − 6 = 0. Factorise: (x − 3)(x + 2) = 0, so x = 3 or x = −2.

展开得 x² + 2x + 1 = 3x + 7。整理得 x² − x − 6 = 0。因式分解得 (x − 3)(x + 2) = 0，因此 x = 3 或 x = −2。

11. Common Mistakes to Avoid | 应避免的常见错误

Here are typical pitfalls in quadratic equation problems:

以下是二次方程问题中常见的陷阱：

Forgetting to rearrange the equation to zero before factorising.
在因式分解前忘记把方程化为零的形式。
Misusing the formula by confusing the signs of b and c.
代入公式时弄错 b 和 c 的符号。
Dividing both sides by x when x = 0 is a possible root.
当 x = 0 可能是根时，两边直接除以 x。
Ignoring a negative root in word problems when the context requires a positive answer.
在应用题中忽略负根，而实际情境需要正数答案。
Making arithmetic errors when completing the square, especially with fractions.
配方法中出现算术错误，特别是涉及分数时。

12. Summary and Exam Tips | 总结与考试技巧

For IGCSE Mathematics, always check whether the equation can be factorised first. If not, use the quadratic formula or completing the square. Remember to write your answers to the required degree of accuracy, and always check your solutions by substituting them back.

在IGCSE数学中，先检查方程能否因式分解。如果不能，就使用求根公式或配方法。记住按题目要求写出准确度，并且通过代入检查解。

Key formulas to memorise:

需要牢记的关键公式：

Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)

求根公式：x = (−b ± √(b² − 4ac)) / (2a)

Axis of symmetry: x = −b / (2a)

对称轴：x = −b / (2a)

Practise a variety of problems until the methods become automatic. This is one of the most rewarding topics for gaining marks in the exam.

多练习不同类型的题目，直到方法运用自如。这是考试中非常容易得分的主题之一。

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Quadratic Equations & Their Solutions | IGCSE数学：二次方程及其解法