📚 Mastering Quadratic Equations for Cambridge IGCSE Mathematics | 攻克剑桥 IGCSE 数学二次方程
Quadratic equations appear throughout the Cambridge IGCSE Mathematics syllabus, from basic factorising in Core to discriminant analysis and graphical transformations in Extended. This revision guide explains the essential methods, common pitfalls, and exam-ready strategies in clear steps.
二次方程贯穿剑桥 IGCSE 数学课程，从 Core 的基础因式分解到 Extended 的判别式分析和图像变换。本复习指南以清晰的步骤讲解核心方法、常见易错点以及备考策略。
1. What Is a Quadratic Equation? | 什么是二次方程？
d a ≠ 0. The highest power of the variable x is 2, which gives the equation its characteristic U-shaped graph called a parabola.
二次方程是任何可以写成 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，这使得它的图像呈特有的 U 形，称为抛物线。
ial. Quadratic equations can have two real roots, one repeated real root, or no real roots depending on the discriminant.
如果 a = 0，方程就变成线性方程 bx + c = 0，因此条件 a ≠ 0 至关重要。根据判别式，二次方程可能有两个实根、一个重根或没有实根。
In Core examinations, you will usually solve quadratics with integer roots. Extended papers may include surds, decimals, and problems that require interpreting the discriminant.
在 Core 考试中，通常解根为整数的二次方程。Extended 试卷可能包含根为无理数、小数以及需要解释判别式的问题。
2. Standard Form and Key Terminology | 标准形式与关键术语
a is called the leading coefficient, b is the linear coefficient, and c is the constant term. Identifying these values correctly is the first step in most solution methods.
二次方程的标准形式为 ax² + bx + c = 0。系数 a 称为二次项系数，b 为一次项系数，c 为常数项。正确识别这些值是大多数解题方法的第一步。
, and c = 3. If the equation is not in standard form, collect all terms on one side first.
例如，在 2x² − 5x + 3 = 0 中，a = 2，b = −5，c = 3。如果方程不是标准形式，应先把所有项移到一边。
- Leading coefficient a must not be zero.
- 二次项系数 a 不能为零。
- Roots are the x-values that satisfy the equation.
- 根是使方程成立的 x 值。
- The y-intercept of the graph is (0, c).
- 图像的 y 轴截距为 (0, c)。
3. Solving by Factorising | 因式分解法
Factorising is usually the fastest method when the quadratic can be written as a product of two linear factors. For x² + bx + c, find two numbers that multiply to c and add to b.
当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。对于 x² + bx + c，找到两个数，它们的积为 c，和为 b。
fore x² + 5x + 6 = (x + 2)(x + 3) = 0. Setting each factor to zero gives x = −2 or x = −3.
例如：解 x² + 5x + 6 = 0。我们需要两个数，积为 6 且和为 5：2 和 3。因此 x² + 5x + 6 = (x + 2)(x + 3) = 0。令每个因式为零，得 x = −2 或 x = −3。
or trial and error. For example, 2x² + 7x + 3 = (2x + 1)(x + 3), giving x = −½ or x = −3.
当 a ≠ 1 时，可使用拆中项法或尝试法。例如，2x² + 7x + 3 = (2x + 1)(x + 3)，得 x = −½ 或 x = −3。
isation by expanding: (x + 2)(x + 3) = x² + 5x + 6, which matches the original expression.
务必通过展开来检查因式分解：(x + 2)(x + 3) = x² + 5x + 6，与原式一致。
4. Solving by Completing the Square | 配方法
b/2)². This method is essential for deriving the quadratic formula and for finding turning points.
配方法将 x² + bx 转化为完全平方三项式。关键公式是 x² + bx = (x + b/2)² − (b/2)²。此方法是推导二次公式和求顶点的基础。
x² + 6x = (x + 3)² − 9, so x² + 6x + 2 = (x + 3)² − 7 = 0. Then (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7.
例如：解 x² + 6x + 2 = 0。首先，x² + 6x = (x + 3)² − 9，所以 x² + 6x + 2 = (x + 3)² − 7 = 0。于是 (x + 3)² = 7，即 x + 3 = ±√7，得 x = −3 ± √7。
, factor out a first: 2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5.
当 a ≠ 1 时，先提取 a：2x² + 8x + 3 = 2(x² + 4x) + 3 = 2[(x + 2)² − 4] + 3 = 2(x + 2)² − 5。
the vertex form y = a(x − h)² + k directly, which is useful for graph sketching questions.
该方法还直接给出顶点式 y = a(x − h)² + k，这对画图像题非常有用。
5. The Quadratic Formula | 二次公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is given by:
二次公式可解任何二次方程 ax² + bx + c = 0。公式如下：
x = (−b ± √b² − 4ac) / (2a)
This formula is derived by completing the square on the general quadratic. It is particularly useful when factorising is difficult or when coefficients are decimals or fractions.
该公式是通过对一般二次式配方推导出来的。当因式分解困难或系数为小数、分数时尤其有用。
Example: Solve 3x² − 4x − 2 = 0. Here a = 3, b = −4, c = −2. Substituting gives x = (4 ± √16 + 24) / 6 = (4 ± √40) / 6 = (4 ± 2√10) / 6 = (2 ± √10) / 3.
例如：解 3x² − 4x − 2 = 0。这里 a = 3，b = −4，c = −2。代入得 x = (4 ± √16 + 24) / 6 = (4 ± √40) / 6 = (4 ± 2√10) / 6 = (2 ± √10) / 3。
Remember to put brackets around the numerator when using a calculator, and simplify surds fully to gain all accuracy marks.
使用计算器时记得给分子加上括号，并完全化简根式以获得所有准确性分数。
6. Discriminant and Types of Roots | 判别式与根的类型
c formula: Δ = b² − 4ac. It tells us the nature of the roots without solving the equation.
判别式是二次公式中平方根内的表达式：Δ = b² − 4ac。它无需解方程即可判断根的性质。
- If Δ > 0, there are two distinct real roots.
- 如果 Δ > 0，则有两个不相等的实根。
- If Δ = 0, there is one repeated real root.
- 如果 Δ = 0，则有一个重根。
- If Δ < 0, there are no real roots (the roots are complex, but not needed for IGCSE).
- 如果 Δ < 0，则没有实根（根为复数，IGCSE 不要求）。
r x² − 6x + 9 = 0, Δ = (−6)² − 4(1)(9) = 36 − 36 = 0, so there is one repeated root x = 3.
例如：对于 x² − 6x + 9 = 0，Δ = (−6)² − 4(1)(9) = 36 − 36 = 0，因此有一个重根 x = 3。
= 0 has real roots. This requires setting the discriminant greater than or equal to zero.
在 Extended 试卷中，可能会要求找出使 kx² + 2x + 1 = 0 有实根的 k 的取值范围。这需要令判别式大于或等于零。
7. Graphs of Quadratic Functions | 二次函数图像
. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上，有最小值点；如果 a < 0，开口向下，有最大值点。
of ax² + bx + c = 0. If the discriminant is negative, the graph does not cross the x-axis.
y 轴截距为 (0, c)。x 轴截距是方程 ax² + bx + c = 0 的实根。如果判别式为负，图像不与 x 轴相交。
intercept (0, 3) and roots x = 1 and x = 3, so it crosses the x-axis at (1, 0) and (3, 0).
例如：y = x² − 4x + 3 的 y 轴截距为 (0, 3)，根为 x = 1 和 x = 3，因此图像在 (1, 0) 和 (3, 0) 处与 x 轴相交。
Sketching questions often require you to label intercepts, the turning point, and the axis of symmetry clearly.
画图题通常要求清晰标出截距、顶点和对称轴。
8. Turning Point and Symmetry | 顶点与对称性
e x-coordinate of the vertex is given by x = −b / (2a). This is also the axis of symmetry.
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