📚 Quadratic Equations: Factoring, Formula & Graphs | 二次方程：因式分解、公式与图像
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, coordinate geometry, measurement problems, and even some trigonometry questions. Understanding how to solve, factor, and sketch quadratics gives you a strong toolkit for Paper 2 and Paper 4, whether you are taking Core or Extended. This article reviews the key methods and common exam traps in a clear, bilingual format.
二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、坐标几何、测量问题，甚至一些三角学题目中。理解如何求解、因式分解并绘制二次函数图像，能为你参加 Paper 2 和 Paper 4 考试提供一套强大的解题工具，无论你参加的是 Core 还是 Extended。本文以清晰的双语形式回顾主要方法和常见考试陷阱。
1. What Is a Quadratic Equation? | 什么是二次方程？
stants and a is not zero. If a were zero, the equation would become linear, not quadratic.
二次方程是未知数的最高次数为 2 的方程。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 是常数，并且 a 不等于 0。如果 a 等于 0，方程就变成一次方程，而不是二次方程。
Quadratic equations can have two real solutions, one repeated real solution, or no real solutions. The exact number depends on the discriminant, which is studied in a later section. In IGCSE, you need to know how to solve quadratics by factorising, by using the quadratic formula, and by completing the square.
二次方程可以有两个实数解、一个重复的实数解，或者没有实数解。具体的解的数量取决于判别式，这一点将在后面小节中学习。在 IGCSE 中，你需要掌握因式分解法、公式法以及配方法求解二次方程。
2. Standard Form and Key Terms | 标准形式与关键术语
The standard form of a quadratic equation is written with all terms on one side and zero on the other side:
二次方程的标准形式是将所有项放在一边，另一边为 0：
ax² + bx + c = 0, a ≠ 0
Here, the term ax² is called the quadratic term, bx is the linear term, and c is the constant term. The coefficient a is the leading coefficient and controls the direction and width of the parabola when the equation is graphed.
这里，ax² 称为二次项，bx 称为一次项，c 称为常数项。系数 a 是首项系数，当方程用图像表示时，它控制抛物线的开口方向和宽度。
come 3x² − 7x + 2 = 0. This avoids sign errors in factorising or in the quadratic formula.
在应用任何方法之前，务必先将方程整理为标准形式。例如，3x² = 7x − 2 应首先化为 3x² − 7x + 2 = 0。这样可以避免因式分解或代入二次公式时出现符号错误。
3. Solving by Factorising | 因式分解法求解
Factorising is often the fastest method when the quadratic has simple integer roots. The idea is to write ax² + bx + c as a product of two linear brackets, then set each bracket equal to zero.
当二次方程的根为简单整数时，因式分解通常是最快的方法。其思路是将 ax² + bx + c 写成两个一次括号的乘积，然后令每个括号等于 0。
and −3, so x² − 5x + 6 = (x − 2)(x − 3). Setting each factor to zero gives x = 2 or x = 3.
例如：解 x² − 5x + 6 = 0。我们需要找到两个数，它们的乘积为 +6，和为 −5。这两个数是 −2 和 −3，所以 x² − 5x + 6 = (x − 2)(x − 3)。令每个因式等于 0，得 x = 2 或 x = 3。
+ 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1). The solutions are x = −3/2 and x = −1.
当 x² 的系数不是 1 时，因式分解需要更加仔细。对于 2x² + 5x + 3，可以拆分中间项：2x² + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1)。解为 x = −3/2 和 x = −1。
Always check your factorisation by expanding the brackets. This small habit prevents many careless mistakes in exams.
务必通过展开括号来检验因式分解是否正确。这个小习惯可以避免考试中的许多粗心错误。
4. Solving by Completing the Square | 配方法求解
Completing the square is useful when a quadratic cannot be easily factorised. It is also the method that leads to the quadratic formula. The goal is to rewrite the quadratic expression in the form (x + p)² + q.
当二次方程不易因式分解时，配方法非常有用。这个方法也引出了二次公式。其目标是将二次式改写为 (x + p)² + q 的形式。
. Write x² + 6x + 9 − 9 + 2 = 0, so (x + 3)² − 7 = 0. Therefore (x + 3)² = 7, and x = −3 ± √7.
对于 x² + 6x + 2 = 0，取 x 的系数 6 的一半，即 3，然后平方得到 9。写成 x² + 6x + 9 − 9 + 2 = 0，所以 (x + 3)² − 7 = 0。因此 (x + 3)² = 7，x = −3 ± √7。
If the coefficient of x² is not 1, divide the whole equation by that coefficient first. Completing the square also gives the vertex form of a quadratic function, which helps identify the turning point of a parabola.
如果 x² 的系数不是 1，首先将整个方程除以该系数。配方法还能给出二次函数的顶点式，有助于确定抛物线的顶点。
(x + p)² = q ⇒ x = −p ± √q
In exams, completing the square is often tested in the Extended paper, but Core students should still recognise the vertex form.
在考试中，配方法常出现在 Extended 试卷中，但 Core 学生仍应认识顶点式。
5. The Quadratic Formula | 二次公式法
is especially useful when factorising is difficult or when coefficients are not integers.
二次公式可以求出任意二次方程 ax² + bx + c = 0 的解。当因式分解困难或系数不是整数时，它尤其有用。
x = (−b ± √b² − 4ac) / (2a)
. For example, in 2x² − 4x − 6 = 0, a = 2, b = −4, and c = −6. Substituting gives x = (4 ± √16 + 48) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4, so x = 3 or x = −1.
使用公式时，首先将方程写成标准形式。然后正确识别 a、b、c。例如，在 2x² − 4x − 6 = 0 中，a = 2，b = −4，c = −6。代入后得到 x = (4 ± √16 + 48) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4，所以 x = 3 或 x = −1。
b = −4, the formula uses −b = 4. Always use brackets when substituting into a calculator.
注意 b 为负数的情况。一个常见错误是当 b 已经是负数时仍把 −b 写成负数。对于 b = −4，公式中应使用 −b = 4。在计算器代入时务必使用括号。
6. The Discriminant | 判别式
The discriminant of a quadratic equation is the part under the square root in the quadratic formula. It is usually written as Δ or D.
二次方程的判别式是二次公式中根号下的部分。它通常写作 Δ 或 D。
Δ = b² − 4ac
The discriminant tells you how many real roots the equation has before you solve it. The three cases are shown below.
判别式可以在求解之前告诉你方程有多少个实数根。以下为三种情况。
| Discriminant | 判别式 | Number of real roots | 实数根数量 |
| Δ > 0 | Two distinct real roots | 两个不同的实数根 |
| Δ = 0 | One repeated real root | 一个重复的实数根 |
| Δ < 0 | No real roots | 没有实数根 |
tic with Δ < 0 may still have complex roots, but these are not required at IGCSE level.
例如，x² − 4x + 4 = 0 的判别式 Δ = (−4)² − 4(1)(4) = 16 − 16 = 0，因此它有一个重复根 x = 2。判别式小于 0 的二次方程可能有复数根，但 IGCSE 不要求掌握复数根。
7. Graphs of Quadratic Functions | 二次函数图像
+ c has a graph called a parabola. If a > 0, the parabola opens upward and looks like a 'U'. If a < 0, it opens downward and looks like an 'n'.
二次函数 y = ax² + bx + c 的图像称为抛物线。如果 a > 0，抛物线开口向上，呈“U”形。如果 a < 0，抛物线开口向下，呈“n”形。
The vertex is the turning point of the parabola. Its x-coordinate is given by −b/(2a). You can substitute this x-value back into the equation to find the y-coordinate of the vertex.
顶点是抛物线的转折点。其 x 坐标由 −b/(2a) 给出。你可以将这个 x 值代回方程，求出顶点的 y 坐标。
x = −b/(2a) and y = f(−b/(2a))
tion is x = −b/(2a). The y-intercept is found by setting x = 0, so it is always at (0, c).
对称轴是经过顶点的垂直线。其方程为 x = −b/(2a)。y 轴截距通过令 x = 0 求得，因此它总是在 (0, c) 处。
Sketching a quadratic accurately means showing the correct shape, the y-intercept, the roots if they exist, and the coordinates of the vertex. Label these features clearly.
准确绘制二次函数图像意味着要画出正确的形状、y 轴截距、存在的根以及顶点坐标。应清楚地标出这些特征。
8. Roots and Intercepts | 根与截距
= 0. Therefore solving the equation is the same as finding the x-intercepts of the graph.
二次方程 ax² + bx + c = 0 的根是 y = ax² + bx + c 的图像与 x 轴相交处的 x 值。在这些点处 y = 0。因此求解方程就等同于找出图像的 x 轴截距。
es the x-axis at the vertex. If Δ < 0, the graph does not cross the x-axis at all. This visual link is often tested in graph-sketching questions.
如果 Δ > 0，图像与 x 轴相交两次。如果 Δ = 0，图像在顶点处刚好接触 x 轴。如果 Δ < 0，图像与 x 轴完全不相交。这一图形联系经常在画图题中考查。
g x = 0. For y = x² − 3x + 2, the y-intercept is (0, 2) and the roots are x = 1 and x = 2.
y 轴截距与根不同。它是图像与 y 轴相交的点，通过代入 x = 0 求得。对于 y = x² − 3x + 2，y 轴截距为 (0, 2)，根为 x = 1 和 x = 2。
9. Applications and Word Problems | 应用与文字题
Quadratic equations are used to model many real-life problems such as area, projectile motion, and product relationships. In word problems, define the unknown with a variable, write an equation from the given information, solve it, and then check whether the answer makes sense in the original context.
二次方程可用于建立许多实际问题的模型，例如面积、抛体运动以及乘积关系。在文字题中，用变量设未知数，根据给定信息写出方程，求解后检查答案在原情境中是否合理。
x = −8 or x = 5. Since width cannot be negative, the width is 5 cm and the length is 8 cm.
例如：一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x cm，则长为 x + 3 cm。面积方程为 x(x + 3) = 40，化为 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。因为宽不能为负数，所以宽为 5 cm，长为 8 cm。
Always reject solutions that do not fit the physical situation, such as negative lengths or times. Also remember to give the final answer with correct units.
务必舍去不符合实际情境的解，例如负的长度或时间。同时记住最终答案要带正确的单位。
10. Common Mistakes and Exam Tips | 常见错误与备考建议
x − 2 = 0 and x + 3 = 0 must be considered. Writing only x = 2 loses the solution x = −3.
一个常见错误是因式分解后忘记令每个因式等于 0。对于 (x − 2)(x + 3) = 0，必须同时考虑 x − 2 = 0 和 x + 3 = 0。只写 x = 2 会丢失解 x = −3。
c must come from ax² + bx + c = 0, not from a rearranged or partially rearranged equation.
另一个错误是在将方程化为标准形式之前就使用二次公式。a、b、c 的值必须来自 ax² + bx + c = 0，而不是来自未整理或部分整理的方程。
When sketching graphs, students sometimes draw a straight line instead of a smooth curve, or they forget the line of symmetry. Use a table of values if necessary, and check the direction of opening from the sign of a.
绘制图像时，学生有时会画成直线而不是平滑曲线，或者忘记对称轴。如有必要可使用数值表，并根据 a 的符号检查开口方向。
Practice past paper questions under timed conditions. Show your working clearly, because method marks are awarded even if the final answer is wrong. A strong grasp of quadratics will also support later topics like simultaneous equations, functions, and coordinate geometry.
在计时条件下练习历年真题。清晰地展示解题过程，因为即使最终答案错误，方法步骤也能得分。扎实掌握二次方程还能为后续的联立方程组、函数和坐标几何等主题提供支持。
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