📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学中的二次方程
Quadratic equations appear in nearly every IGCSE Mathematics paper, both as standalone algebra questions and as part of wider problem-solving tasks. This article covers the core methods, graphs, discriminant, and common exam traps you need to master.
二次方程几乎出现在每一份 IGCSE 数学试卷中，既有独立的代数题，也常作为综合应用题的一部分出现。本文将涵盖核心解法、图像、判别式以及常见的考试陷阱，帮助你全面掌握这一考点。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation in which the highest power of the variable is 2. It generally involves an x² term, an x term, and a constant term.
二次方程是未知数最高次数为 2 的多项式方程。它通常包含 x² 项、x 项和常数项。
The standard form of a quadratic equation is:
二次方程的标准形式为：
ax² + bx + c = 0
he equation becomes linear, not quadratic, so checking this condition is always important.
其中 a、b、c 为常数，且 a 不能等于 0。如果 a = 0，方程就变为一次方程而不是二次方程，因此始终要确认这一条件。
r, such as 3x² − 7 = 2x, and your first step should be to rearrange it into standard form.
在 IGCSE 题目中，二次方程可能会以略微不同的顺序给出，例如 3x² − 7 = 2x，你的第一步应是将其整理成标准形式。
2. Standard Form and Key Vocabulary | 标准形式与关键术语
In the expression ax² + bx + c, the term ax² is called the quadratic term, bx is the linear term, and c is the constant term.
在表达式 ax² + bx + c 中，ax² 称为二次项，bx 称为一次项，c 称为常数项。
The coefficient a is the leading coefficient. It determines whether the parabola opens upwards or downwards, and it also affects how narrow or wide the graph appears.
系数 a 是首项系数。它决定抛物线的开口朝上还是朝下，同时也影响图像的宽窄程度。
The solutions of the quadratic equation are called roots. They are the x-values that make the equation true, and on a graph they correspond to the x-intercepts.
二次方程的解称为根。它们是使方程成立的 x 值，在图像上对应的是 x 轴截距。
Using the correct vocabulary helps in exam settings, especially when a question asks for the roots, solutions, or x-intercepts: these all refer to the same values.
使用正确的术语在考试中很有帮助，尤其是当题目问“根”“解”或“x 轴截距”时，它们指的都是同一组数值。
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. You must first rewrite the equation so that one side equals zero.
当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。你必须先把方程改写为一边等于 0 的形式。
t side as (x − 2)(x − 3) = 0. Then set each factor equal to zero to obtain x = 2 or x = 3.
例如，要解 x² − 5x + 6 = 0，将左边因式分解为 (x − 2)(x − 3) = 0。然后令每个因式等于 0，得到 x = 2 或 x = 3。
emove a valid solution. For example, x² = 3x should become x(x − 3) = 0, not simply x = 3.
切勿两边同时除以含有 x 的式子，因为这样可能会丢掉一个有效解。例如 x² = 3x 应化为 x(x − 3) = 0，而不是直接得到 x = 3。
| Quadratic | Factorised form | Solutions |
| x² + 5x + 6 = 0 | (x + 2)(x + 3) = 0 | x = −2 or x = −3 |
| x² − 4x + 4 = 0 | (x − 2)² = 0 | x = 2 (repeated) |
4. Completing the Square | 配方法
Completing the square is a powerful method because it gives both the solutions and the turning point of the quadratic graph. The goal is to rewrite ax² + bx + c in the form a(x − h)² + k.
配方法是一种非常实用的方法，因为它既能求出二次方程的解，也能得到二次函数图像的顶点。其目标是把 ax² + bx + c 化为 a(x − h)² + k 的形式。
en add (6/2)² = 9 to both sides to get x² + 6x + 9 = 4, which is the same as (x + 3)² = 4.
对于 x² + 6x + 5 = 0，先写成 x² + 6x = −5。然后两边同时加上 (6/2)² = 9，得到 x² + 6x + 9 = 4，也就是 (x + 3)² = 4。
(x + 3)² = 4
±2, so x = −1 or x = −5. Remember to include both the positive and negative square roots.
两边开平方得到 x + 3 = ±2，因此 x = −1 或 x = −5。记住要同时写出正负两个平方根。
IGCSE exam papers often ask for the completed square form even when the quadratic can be factorised, so you must be confident with this technique.
IGCSE 试卷经常要求写出配方后的形式，即使该二次式可以因式分解，因此你必须熟练掌握这一技巧。
5. The Quadratic Formula | 二次公式
The quadratic formula can solve any quadratic equation, including those that do not factorise neatly. It is especially useful when the coefficients are large or when the roots are not integers.
二次公式可以求解任何二次方程，包括那些不能整齐因式分解的方程。当系数较大或根不是整数时，二次公式尤其有用。
The formula is:
公式为：
x = [ −b ± √b² − 4ac ] / (2a)
then identify a, b and c carefully. For 2x² + 3x − 2 = 0, we have a = 2, b = 3 and c = −2.
一定要先把二次方程写成标准形式，然后仔细确定 a、b、c。对于 2x² + 3x − 2 = 0，我们有 a = 2，b = 3，c = −2。
Substitute these values into the formula and simplify step by step. Be especially careful with negative coefficients, as subtracting a negative value changes the sign.
将这些值代入公式并逐步化简。要特别注意负系数，因为减去一个负数会改变符号。
In an exam, quote the formula before substituting, and check your final answers by substituting them back into the original equation.
在考试中，应先写出公式再代入数值，并将最终答案代回原方程进行检验。
6. The Discriminant and Number of Roots | 判别式与根的个数
= b² − 4ac. It tells you how many real roots the equation has without actually solving it.
判别式是二次公式中根号下的部分：D = b² − 4ac。它可以在不真正求解方程的情况下判断方程有多少个实根。
D = b² − 4ac
ated real root. If D < 0, there are no real roots, so the graph does not cross the x-axis.
如果 D > 0，方程有两个不相等的实根；如果 D = 0，方程有一个重根；如果 D < 0，方程没有实根，因此图像与 x 轴没有交点。
IGCSE questions often ask you to find the value of k for which a quadratic has equal roots. In that case, set the discriminant equal to zero and solve.
IGCSE 题目经常要求你求出使二次方程具有相等实根的 k 值。此时应令判别式等于 0，然后求解。
| Discriminant D | Number of real roots |
| D > 0 | 2 distinct real roots |
| D = 0 | 1 repeated real root |
| D < 0 | no real roots |
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 的图像是一条抛物线。若 a > 0，抛物线开口向上，具有最低点；若 a < 0，抛物线开口向下，具有最高点。
e x = 0, which gives the point (0, c). The x-intercepts are the roots of ax² + bx + c = 0.
y 轴截距很容易求：代入 x = 0，得到点 (0, c)。x 轴截距就是方程 ax² + bx + c = 0 的根。
When sketching a quadratic graph, always find the y-intercept, the roots if they exist, the axis of symmetry, and the coordinates of the turning point.
画二次函数图像时，始终要求出 y 轴截距、存在的根、对称轴以及顶点坐标。
If the discriminant is negative, the graph does not meet the x-axis, but you can still sketch its shape using the y-intercept and the turning point.
如果判别式为负，图像与 x 轴没有交点，但你仍然可以利用 y 轴截距和顶点画出图像的大致形状。
8. Turning Point and Axis of Symmetry | 顶点与对称轴
The axis of symmetry for a parabola y = ax² + bx + c is a vertical line given by:
抛物线 y = ax² + bx + c 的对称轴是一条垂直直线，其方程为：
x = −b / (2a)
f symmetry. To find its y-coordinate, substitute x = −b / (2a) into the original equation.
顶点位于对称轴上。要求顶点的 y 坐标，可以将 x = −b / (2a) 代入原方程。
e turning point is simply (h, k). This is why completing the square is so useful for graph questions.
如果二次式已经写成配方法形式 y = a(x − h)² + k，那么顶点坐标就是 (h, k)。这就是为什么配方法在图像题中非常有用。
For y = 2(x − 3)² + 1, the turning point is (3, 1) and the axis of symmetry is x = 3.
对于 y = 2(x − 3)² + 1，顶点为 (3, 1)，对称轴为 x = 3。
9. Applications and Real-Life Problems | 实际应用问题
Quadratic equations often appear in real-life contexts such as projectile motion, maximum area, profit modelling, and simple physics problems. The key is to form an equation from the given information.
二次方程经常出现在抛体运动、最大面积、利润建模和简单物理问题等实际情境中。关键是根据已知信息建立方程。
delled by h = −5t² + 20t + 1, the maximum height occurs at t = −20 / (2 × −5) = 2 seconds.
例如，如果球的高度由 h = −5t² + 20t + 1 表示，那么最大高度出现在 t = −20 / (2 × −5) = 2 秒时。
te t = 2 into the equation to find the maximum height: h = −5(2)² + 20(2) + 1 = 21 metres.
将 t = 2 代入方程可求出最大高度：h = −5(2)² + 20(2) + 1 = 21 米。
Always interpret your solutions in the context of the question. A negative time
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