📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程完全指南
Quadratic equations form one of the most important building blocks in IGCSE Mathematics. You will meet them in algebra, in graph sketching, and in real-world problems such as projectile motion, area, and business profit. This article explains the key methods for solving quadratics, how to interpret their graphs, and the common exam errors to avoid.
二次方程是 IGCSE 数学中最重要的基础内容之一。你会在代数、函数图像以及抛体运动、面积和商业利润等实际问题中遇到它们。本文讲解求解二次方程的主要方法、如何理解二次函数图像，以及考试中需要避免的常见错误。
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 standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.
二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。如果 a = 0，方程就变成了一次方程，而不是二次方程。
For example, 2x² - 3x + 1 = 0 is quadratic because the highest power of x is 2. The coefficients are a = 2, b = -3 and c = 1.
例如，2x² - 3x + 1 = 0 是二次方程，因为 x 的最高次数是 2。它的系数为 a = 2、b = -3、c = 1。
In IGCSE exams, you may be asked to solve a quadratic equation, sketch its graph, or use it to model a practical situation. You must therefore be confident in several methods, not just one.
在 IGCSE 考试中，你可能需要解二次方程、画函数图像，或者用它来建立实际情境的数学模型。因此你必须熟练掌握多种方法，而不仅仅是一种。
2. Expanding and Rearranging into Standard Form | 展开并整理为标准形式
Many quadratic problems start with an expression in factorised or disguised form. You need to expand brackets and collect like terms before you can solve the equation. For example, (x + 3)(x - 2) = 0 expands to x² + x - 6 = 0.
许多二次方程问题给出的式子一开始是乘积形式或隐含形式。你需要先展开括号并合并同类项，然后才能求解。例如，(x + 3)(x - 2) = 0 展开后得到 x² + x - 6 = 0。
Rearranging is also useful when an equation is not equal to zero, such as x² + 4x = 5. Subtracting 5 from both sides gives x² + 4x - 5 = 0, which is ready for factorising.
当方程右边不为 0 时，移项整理也很重要，例如 x² + 4x = 5。两边同时减去 5，得到 x² + 4x - 5 = 0，这就为因式分解做好了准备。
Always check that the standard form is ax² + bx + c = 0 before applying factorising, completing the square, or the quadratic formula. This small step prevents sign errors later.
在应用因式分解法、配方法或二次公式之前，一定要先确认方程已经是 ax² + bx + c = 0 的标准形式。这个小小的步骤可以避免后面出现符号错误。
3. Solving by Factorising | 因式分解法
Factorising is usually the fastest method when the quadratic can be written as the product of two linear brackets. The key idea is the zero product property: if (x + p)(x + q) = 0, then either x + p = 0 or x + q = 0.
当二次方程可以写成两个一次括号的乘积时，因式分解通常是最快的方法。其核心思想是零乘积性质：如果 (x + p)(x + q) = 0，那么 x + p = 0 或 x + q = 0。
For example, solve x² + 6x + 8 = 0. We need two numbers that multiply to 8 and add to 6, namely 2 and 4. So x² + 6x + 8 = (x + 2)(x + 4) = 0. The solutions are x = -2 or x = -4.
例如，解方程 x² + 6x + 8 = 0。我们需要找到两个数，它们的乘积为 8，和为 6，这两个数是 2 和 4。所以 x² + 6x + 8 = (x + 2)(x + 4) = 0。解为 x = -2 或 x = -4。
When the leading coefficient is not 1, you may need to factorise by grouping. For instance, 2x² + 5x + 3 becomes (2x + 3)(x + 1). You can check by expanding the brackets.
当二次项系数不是 1 时，你可能需要分组分解。例如，2x² + 5x + 3 可以分解为 (2x + 3)(x + 1)。可以通过展开括号来检验。
Do not forget special cases such as the difference of two squares: x² - 9 = (x + 3)(x - 3). Recognising these patterns saves time in the exam.
不要忘记平方差等特殊形式：x² - 9 = (x + 3)(x - 3)。识别这些模式可以在考试中节省时间。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x - h)² + k, which is especially useful for finding the vertex of a graph and for solving when factorising is difficult.
配方法可以把二次式写成 a(x - h)² + k 的形式，这种形式在求函数图像顶点以及因式分解困难时特别有用。
To complete the square for x² + 6x + 1, start with (x + 3)² = x² + 6x + 9. Since we have +1 instead of +9, subtract 8: x² + 6x + 1 = (x + 3)² - 8.
对 x² + 6x + 1 配方时，先考虑 (x + 3)² = x² + 6x + 9。由于原式是 +1 而不是 +9，因此需要减去 8：x² + 6x + 1 = (x + 3)² - 8。
To solve x² + 6x + 1 = 0, write (x + 3)² - 8 = 0. Then (x + 3)² = 8, so x + 3 = ±√8. The solutions are x = -3 ± 2√2.
解方程 x² + 6x + 1 = 0 时，写成 (x + 3)² - 8 = 0。于是 (x + 3)² = 8，所以 x + 3 = ±√8。解为 x = -3 ± 2√2。
Completing the square also reveals the minimum or maximum value. For y = (x + 3)² - 8, the vertex is (-3, -8) and the graph opens upwards.
配方法还能显示函数的最小值或最大值。对于 y = (x + 3)² - 8，顶点为 (-3, -8)，图像开口向上。
5. The Quadratic Formula | 二次公式
The quadratic formula solves any quadratic equation, even when factorising is not possible. The solutions of ax² + bx + c = 0 are given by:
二次公式可以求解任何二次方程，即使无法因式分解。方程 ax² + bx + c = 0 的解为：
x = (-b ± √b² - 4ac) / 2a
For example, solve 2x² - 4x - 3 = 0. Here a = 2, b = -4 and c = -3. Substituting gives x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4. This simplifies to x = (2 ± √10) / 2.
例如，解方程 2x² - 4x - 3 = 0。这里 a = 2，b = -4，c = -3。代入公式得 x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4。化简为 x = (2 ± √10) / 2。
Always write the formula clearly before substituting values. Be careful with negative signs: if b = -4, then -b = 4.
在代入数值之前，一定要先清楚地写出公式。注意负号：如果 b = -4，那么 -b = 4。
The quadratic formula is also useful for checking solutions obtained by factorising. In the exam, if you have time, using it as a backup can catch arithmetic errors.
二次公式也可以用来检验因式分解法得到的解。在考试中，如果时间允许，用它作为备用方法可以检查出计算错误。
6. The Discriminant and Nature of Roots | 判别式与根的性质
The expression under the square root in the quadratic formula is called the discriminant: Δ = b² - 4ac. It tells us how many real roots the quadratic equation has.
二次公式中根号下的表达式称为判别式：Δ = b² - 4ac。它可以告诉我们二次方程有多少个实数根。
If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root, meaning the graph touches the x-axis at one point. If Δ < 0, there are no real roots, meaning the graph does not cross the x-axis.
如果 Δ > 0，则有两个不相等的实数根。如果 Δ = 0，则有一个重复的实数根，即图像在一点与 x 轴相切。如果 Δ < 0，则没有实数根，即图像不与 x 轴相交。
For x² - 5x + 6 = 0, the discriminant is 25 - 24 = 1, so there are two real roots. For x² - 4x + 4 = 0, the discriminant is 16 - 16 = 0, so there is one repeated root.
对于 x² - 5x + 6 = 0，判别式为 25 - 24 = 1，所以有两个实数根。对于 x² - 4x + 4 = 0，判别式为 16 - 16 = 0，所以有一个重复根。
Exam questions often ask you to find the range of values for which a quadratic has real roots. Set b² - 4ac ≥ 0 and solve the resulting inequality.
考试题目经常要求你求二次方程有实数根时参数的取值范围。设 b² - 4ac ≥ 0，然后解所得不等式。
7. Sketching Quadratic Graphs | 二次函数图像
A quadratic function y = ax² + bx + c produces a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
二次函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，存在最小值点。如果 a < 0，抛物线开口向下，存在最大值点。
To sketch a quadratic graph, find three key features: the y-intercept, the x-intercepts, and the vertex. The y-intercept is simply c, because when x = 0, y = c.
画二次函数图像时，需要找出三个关键特征：y 轴截距、x 轴截距和顶点。y 轴截距就是 c，因为当 x = 0 时，y = c。
The x-intercepts are the solutions of ax² + bx + c = 0. The x-coordinate of the vertex can be found using x = -b / 2a.
x 轴截距是方程 ax² + bx + c = 0 的解。顶点的 x 坐标可以用 x = -b / 2a 求出。
For example, y = x² - 4x + 3 has a y-intercept at (0, 3), x-intercepts at x = 1 and x = 3, and a vertex at (2, -1). Plotting these gives a clear parabola.
例如，y = x² - 4x + 3 的 y 轴截距为 (0, 3)，x 轴截距为 x = 1 和 x = 3，顶点为 (2, -1)。标出这些点就能画出清晰的抛物线。
8. Word Problems and Modelling | 应用题与建模
Quadratic equations often appear in practical problems involving areas, products, and projectile motion. Translate the words into an equation, solve it, and then check that the answer makes sense in the original context.
二次方程经常出现在涉及面积、乘积和抛体运动的实际问题中。先把文字转化为方程，求解后再检查答案在原始情境中是否合理。
For example, a rectangular garden has length 3 m longer than its width. If the area is 40 m², let the width be x. Then the length is x + 3, so x(x + 3) = 40. This rearranges to x² + 3x - 40 = 0.
例如，一个矩形花园的长比宽多 3 米。如果面积为 40 平方米，设宽为 x。那么长为 x + 3，所以 x(x + 3) = 40。整理得 x² + 3x - 40 = 0。
Factorising gives (x + 8)(x - 5) = 0, so x = -8 or x = 5. Since width cannot be negative, the width is 5 m and the length is 8 m.
因式分解得 (x + 8)(x - 5) = 0，所以 x = -8 或 x = 5。由于宽度不能为负数，因此宽为 5 米，长为 8 米。
In projectile problems, the height h after time t may be modelled by a quadratic. Setting h = 0 tells you when the object returns to the ground.
在抛体运动问题中，物体在时间 t 后的高度 h 可以用二次式表示。令 h = 0，就能求出物体何时回到地面。
9. Common Mistakes to Avoid | 常见错误
One common mistake is forgetting to set the equation equal to zero before factorising. If you factorise x² + 4x = 5 without rearranging, you will not be able to use the zero product property correctly.
一个常见错误是在因式分解前没有把方程整理为等于 0 的形式。如果直接对 x² + 4x = 5 进行因式分解，就无法正确使用零乘积性质。
Another error involves sign mistakes when using the quadratic formula. If b = -3, then -b = 3, and b² = 9, not -9.
另一个错误是在使用二次公式时出现符号错误。如果 b = -3，那么 -b = 3，而 b² = 9，不是 -9。
Students also sometimes give only one solution when a quadratic has two. Remember that a quadratic equation can have two, one, or no real solutions, and you must state all possible values of x.
学生有时只写出二次方程的一个解，而它实际有两个解。要记住二次方程可能有两个、一个或没有实数解，你必须写出 x 的所有可能值。
Finally, do not use the quadratic formula with a = 0. If the coefficient of x² is zero, the equation is linear, and you should solve it directly.
最后，当 a = 0 时不要使用二次公式。如果 x² 的系数为 0，这个方程就是一次的，应该直接求解。
10. Exam Tips and Practice
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