📚 Quadratic Equations and Graphs for IGCSE Maths | IGCSE 数学：二次方程与图像
Quadratic equations and their graphs are central to the IGCSE Mathematics syllabus, appearing in algebra, coordinate geometry and applied problem solving. This article brings together the essential methods for solving quadratics, interpreting the discriminant, and sketching parabolas accurately. You will also see how to avoid common errors in exams.
二次方程及其图像是 IGCSE 数学课程的核心内容，广泛出现在代数、坐标几何和应用题中。本文将汇总求解二次方程的主要方法、判别式的意义以及准确绘制抛物线图像的关键技巧。同时还会说明如何避免考试中的常见错误。
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 general form below, where a, b and c are constants and a is not equal to zero.
二次方程是指可以写成下面一般形式的方程，其中 a、b、c 为常数，且 a 不等于零。
a x² + b x + c = 0, a ≠ 0
Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. The highest power of x is 2, which is why the equation is called quadratic.
这里 a 是 x² 的系数，b 是 x 的系数，c 是常数项。x 的最高次数是 2，因此这类方程被称为二次方程。
ro, the equation would become linear, such as b x + c = 0, so the x² term would disappear.
a ≠ 0 这个条件至关重要。如果 a 为零，方程就会退化为一元一次方程，例如 b x + c = 0，x² 项就消失了。
The roots of a quadratic equation are the values of x that make the equation true. Roots are also called solutions or zeros of the corresponding quadratic function.
二次方程的根是使方程成立的 x 值。根也被称为方程的解，或对应二次函数的零点。
A quadratic equation can have two distinct real roots, one repeated real root, or no real roots. The number and type of roots depend on the discriminant, which we will study later.
二次方程可能有两个不同的实数根、一个重复实数根，或没有实数根。根的个数和类型取决于判别式，后文会详细讨论。
2. Solving by Factorising | 用因式分解法求解
Factorising is often the quickest method when the quadratic expression factorises nicely. It uses the fact that if the product of two factors is zero, then at least one factor must be zero.
当二次式容易分解时，因式分解通常是最快的方法。它利用了一个事实：如果两个因式的乘积为零，那么至少有一个因式为零。
factorises as (x − 3)(x − 4) = 0. Therefore x − 3 = 0 or x − 4 = 0, giving x = 3 or x = 4.
例如，x² − 7x + 12 = 0 可分解为 (x − 3)(x − 4) = 0。因此 x − 3 = 0 或 x − 4 = 0，得到 x = 3 或 x = 4。
common factor first. For example, 3x² − 12x = 0 becomes 3x(x − 4) = 0, so x = 0 or x = 4.
首先要寻找公因式。例如，3x² − 12x = 0 可写成 3x(x − 4) = 0，所以 x = 0 或 x = 4。
ares is another useful pattern: x² − 9 = 0 becomes (x − 3)(x + 3) = 0, so x = 3 or x = −3.
平方差是另一个常用模式：x² − 9 = 0 可写成 (x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。
If a quadratic does not factorise with integers, do not force it. Use completing the square or the quadratic formula instead.
如果二次式不能顺利分解为整数因式，不要强行分解。应改用配方法或求根公式。
3. Completing the Square | 配方法
= q. This method is useful for solving equations and for finding the vertex of a parabola.
配方法把二次式改写成 (x + p)² = q 的形式。这个方法既可用于解方程，也可用于求抛物线的顶点。
write x² + 6x = (x + 3)² − 9. The equation becomes (x + 3)² − 9 + 5 = 0, so (x + 3)² = 4.
对于 x² + 6x + 5 = 0，6 的一半是 3，所以 x² + 6x = (x + 3)² − 9。方程变为 (x + 3)² − 9 + 5 = 0，因此 (x + 3)² = 4。
1 or x = −5. Remember that both the positive and negative square roots must be considered.
开平方得到 x + 3 = ±2，所以 x = −1 或 x = −5。要记住正负两个平方根都要考虑。
In general, x² + b x + c can be written as (x + b/2)² + c − (b/2)². This form helps you read the minimum or maximum value directly.
一般来说，x² + b x + c 可以写成 (x + b/2)² + c − (b/2)²。这个形式有助于直接读出最小值或最大值。
+ k shows that the vertex is at (h, k). If a is positive, k is the minimum y-value; if a is negative, k is the maximum y-value.
配方后的形式 y = a(x − h)² + k 表明顶点为 (h, k)。如果 a 为正，k 是最小 y 值；如果 a 为负，k 是最大 y 值。
4. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation, even when factorising is difficult or impossible with simple integers. It is derived by completing the square on the general form.
求根公式可以解任何二次方程，即使因式分解很困难或无法用简单整数分解时也适用。它是对一般形式配方后推导出来的。
x = (−b ± √b² − 4ac) / (2a)
a x² + b x + c = 0, identify a, b and c carefully, then substitute them into the formula.
使用时要先把方程写成标准形式 a x² + b x + c = 0，仔细辨认 a、b、c，然后代入公式。
For example, in 2x² + 3x − 2 = 0, a = 2, b = 3 and c = −2. Substitution gives x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4, so x = 1/2 or x = −2.
例如，在 2x² + 3x − 2 = 0 中，a = 2，b = 3，c = −2。代入得 x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4，所以 x = 1/2 或 x = −2。
Be careful with signs. If b is negative, then −b becomes positive. If c is negative, then −4ac may become positive.
要特别注意符号。如果 b 为负，−b 就为正。如果 c 为负，−4ac 可能变为正。
The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many real roots the equation has before you finish the calculation.
根号下的表达式 b² − 4ac 称为判别式。在完成计算之前，它就能告诉你方程有多少个实数根。
5. The Discriminant and Nature of Roots | 判别式与根的性质
4ac. Its sign determines whether the quadratic equation has two distinct real roots, one repeated real root, or no real roots.
判别式定义为 Δ = b² − 4ac。它的符号决定二次方程有两个不同实根、一个重复实根，还是没有实根。
| Discriminant 判别式 | Nature of roots 根的性质 | Graph 图像 |
|---|---|---|
| Δ > 0 | Two distinct real roots 两个不同实根 | Cuts x-axis twice 与 x 轴有两个交点 |
| Δ = 0 | One repeated real root 一个重复实根 | Touches x-axis once 与 x 轴相切于一点 |
| Δ < 0 | No real roots 无实根 | Does not meet x-axis 与 x 轴无交点 |
= 16 − 16 = 0, so it has one repeated root, x = 2. The graph touches the x-axis at x = 2.
例如，x² − 4x + 4 = 0 的 Δ = 16 − 16 = 0，因此有一个重复根 x = 2。图像在 x = 2 处与 x 轴相切。
arabola. A tangent occurs when the resulting quadratic has exactly one solution, so Δ = 0.
判别式还用于证明一条直线与抛物线相切。相切时，联立得到的二次方程只有一个解，所以 Δ = 0。
If roots exist, their sum is −b/a and their product is c/a. These relationships can help check answers and solve symmetric problems.
如果根存在，它们的和为 −b/a，积为 c/a。这些关系有助于检查答案和解决对称性问题。
6. Graphs of Quadratic Functions: Parabolas | 二次函数图像：抛物线
+ c is a parabola. Every quadratic graph has a U-shaped or an upside-down U-shaped curve.
y = a x² + b x + c 的图像是一条抛物线。每个二次函数的图像都是 U 形或倒 U 形的曲线。
If a > 0, the parabola opens upward and has a minimum point. If a < 0, the parabola opens downward and has a maximum point.
如果 a > 0，抛物线开口向上，有最低点。如果 a < 0，抛物线开口向下，有最高点。
The value of |a| affects the width of the parabola. A larger |a| makes the curve narrower, while a smaller |a| makes it wider.
|a| 的大小影响抛物线的宽窄。|a| 越大，曲线越窄；|a| 越小，曲线越宽。
by setting x = 0. It is always (0, c), where c is the constant term in the standard form.
令 x = 0 可求 y 轴截距，它总是 (0, c)，其中 c 是标准形式中的常数项。
the equation a x² + b x + c = 0. They show where the graph crosses or touches the x-axis.
如果存在 x 轴截距，它们就是方程 a x² + b x + c = 0 的实数根。它们表示图像与 x 轴相交或相切的位置。
7. Key Features: Vertex, Intercepts and Symmetry | 关键特征：顶点、截距与对称性
/(2a). It passes through the vertex and divides the parabola into two mirror-image halves.
对称轴是竖直线 x = −b/(2a)。它穿过顶点，把抛物线分成两个镜像对称的部分。
The vertex lies on the axis of symmetry. Its x-coordinate is −b/(2a), and its y-coordinate is found by substituting this x-value into the equation.
顶点位于对称轴上。它的 x 坐标是 −b/(2a)，y 坐标可通过把这个 x 值代入方程求得。
+ k gives the vertex directly as (h, k). This is often faster than using −b/(2a) when the equation is already in that form.
配方形式 y = a(x − h)²
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