📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, graphs, problem solving, and even in geometry and mensuration. This revision guide covers the key methods, graph properties, and exam strategies you need to master.
二次方程是 IGCSE 数学大纲中最重要的主题之一。它们出现在代数、图像、问题解决，甚至在几何和测量中。本复习指南涵盖你需要掌握的关键方法、图像性质和考试策略。
Whether you are studying the Core or Extended tier, you must be able to solve quadratic equations by factorising, using the quadratic formula, and completing the square. You also need to sketch parabolas, interpret the discriminant, and solve quadratic inequalities.
无论你学习的是核心课程还是扩展课程，你都必须能够通过因式分解、求根公式和配方法解二次方程。你还需要画出抛物线、解释判别式，并解二次不等式。
This article moves from the definition of a quadratic, through each solution method, to graph sketching and exam technique. Every section is written in English and Chinese so that you can revise the mathematics and the language of the exam at the same time.
本文从二次方程的定义出发，依次讲解每一种解法，再到图像绘制和考试技巧。每一节都采用英文与中文对照，让你在复习数学的同时也熟悉考试语言。
1. What is a Quadratic Equation? | 什么是二次方程？
are constants and a ≠ 0. If a were 0, the equation would become linear rather than quadratic. The constants a, b and c are usually called coefficients, and c is sometimes called the constant term.
二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数且 a ≠ 0。如果 a 为 0，方程将变为一次方程而不是二次方程。常数 a、b 和 c 通常称为系数，其中 c 有时被称为常数项。
ax² + bx + c = 0, a ≠ 0
The highest power of the unknown x is 2, which is why these equations are called 'quadratic' oots. These roots are the x-values where the graph of y = ax² + bx + c crosses the x-axis.
未知数 x 的最高次数是 2，这就是这些方程被称为 "二次" 的原因。一个二次方程最多可以有两个实数解，称为根。这些根是函数 y = ax² + bx + c 的图像与 x 轴相交的 x 值。
ays identify the values of a, b and c before choosing a solution method. This small step prevents many careless mistakes.
例如，x² + 5x + 6 = 0 和 2x² - 3x + 1 = 0 都是二次方程。在选择解法之前，一定要先确定 a、b 和 c 的值。这个小步骤可以防止许多粗心错误。
, a = 1, b = -4 and c = -12. Keeping the signs correct is essential in every method below.
请记住，a、b 和 c 可以为负也可以为正。对于方程 x² - 4x - 12 = 0，a = 1，b = -4，c = -12。在下面每一种方法中，保持符号正确都至关重要。
irst rearrange it to 3x² - 5x - 2 = 0 before factorising or substituting into the formula.
在使用任何方法之前，方程必须化为一边为零的标准形式。如果题目给出 3x² = 5x + 2，请先将其整理为 3x² - 5x - 2 = 0，然后再进行因式分解或代入公式。
2. Solving by Factorising | 因式分解法
he form ax² + bx + c = 0, then look for two numbers that multiply to give ac and add to give b.
当二次表达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。将方程写成 ax² + bx + c = 0 的形式，然后寻找两个数，它们的乘积为 ac，和为 b。
)(x - 3) = 0. By the zero product rule, if a product equals zero then at least one factor must be zero, so x = 2 or x = 3.
举一个简单的例子，解方程 x² - 5x + 6 = 0。寻找两个数，它们的乘积为 6，和为 -5。这两个数是 -2 和 -3，因此表达式分解为 (x - 2)(x - 3) = 0。根据零积法则，如果乘积等于零，那么至少有一个因式为零，所以 x = 2 或 x = 3。
if pq = 0 then p = 0 or q = 0
ctorise in pairs: 2x(x - 1) - 1(x - 1) = 0, giving (2x - 1)(x - 1) = 0, so x = ½ or x = 1.
当 a 不为 1 时，同样的规则仍然适用。对于 2x² - 3x + 1 = 0，ac = 2，b = -3，所以两个数是 -1 和 -2。拆分中间项：2x² - 2x - x + 1 = 0。分组分解：2x(x - 1) - 1(x - 1) = 0，得到 (2x - 1)(x - 1) = 0，所以 x = ½ 或 x = 1。
4. Notice that one root is zero here, which is perfectly normal and must not be discarded.
一定要先提取公因式。在 3x² - 12x = 0 中，公因式是 3x，所以 3x(x - 4) = 0，因此 x = 0 或 x = 4。注意这里有一个根是零，这完全正常，绝不能把它舍去。
= -3. Similarly 4x² - 25 = (2x - 5)(2x + 5). Spotting this pattern saves time in the exam.
要能识别平方差公式：x² - 9 = (x - 3)(x + 3)，所以 x² - 9 = 0 给出 x = 3 或 x = -3。同样地，4x² - 25 = (2x - 5)(2x + 5)。识别这一模式可以节省考试时间。
If no pair of integers works, the expression does not factorise nicely. In that case move on immediately to completing the square or the quadratic formula rather than losing minutes guessing.
如果找不到任何一对整数满足条件，说明该表达式无法整齐地分解。此时应立即改用配方法或求根公式，而不是把时间浪费在猜测上。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² + q, which reveals the turning point of the graph directly. This method works for every quadratic, even when the expression will not factorise with integers.
配方法将二次式改写为 (x + p)² + q 的形式，可以直接看出图像的转折点。这种方法适用于所有二次式，即使该表达式无法用整数分解也一样。
x + b/2)² - (b/2)² + c. You take half of the coefficient of x, square the bracket, then subtract the square you have added.
当 a = 1 时的关键恒等式是 x² + bx + c = (x + b/2)² - (b/2)² + c。取 x 的系数的一半，写成平方的括号，再减去多加上的那个平方。
x² + bx + c = (x + b/2)² - (b/2)² + c
- 3)² - 9 + 2 = 0, that is (x - 3)² - 7 = 0. Rearranging gives (x - 3)² = 7, so x - 3 = ±√7 and x = 3 ± √7. Leaving the answer in surd form is usually required unless the question asks for a decimal.
例如，x² - 6x + 2 = 0 变为 (x - 3)² - 9 + 2 = 0，即 (x - 3)² - 7 = 0。整理得 (x - 3)² = 7，所以 x - 3 = ±√7，x = 3 ± √7。除非题目要求小数，通常答案需保留根号形式。
2(x + 2)² - 8 + 3 = 2(x + 2)² - 5. The minimum value is therefore -5, occurring at x = -2.
当 a 不为 1 时，先从前面两项中提出 a。对于 2x² + 8x + 3，写成 2(x² + 4x) + 3 = 2[(x + 2)² - 4] + 3 = 2(x + 2)² - 8 + 3 = 2(x + 2)² - 5。因此最小值为 -5，在 x = -2 处取得。
urning point. The completed square form y = a(x - h)² + k tells you that the vertex is at (h, k).
这就是为什么当题目要求最小值、最大值或转折点坐标时，配方法是最佳选择。配方后的形式 y = a(x - h)² + k 告诉你顶点坐标为 (h, k)。
A useful check: expand your completed square answer and confirm that you recover the original expression. This takes ten seconds and catches sign errors immediately.
一个有用的检验方法：把配方后的结果展开，确认你能还原原来的表达式。这只花十秒钟，却能立刻发现符号错误。
4. The Quadratic Formula | 求根公式
factors. It is obtained by completing the square on the general equation ax² + bx + c = 0.
求根公式可以解任何二次方程，无需猜测因式。它是通过对一般式 ax² + bx + c = 0 配方得到的。
x = (-b ± √b² - 4ac) / (2a)
(3 ± 1) / 4, giving x = 1 or x = ½, which agrees with the factorising result in Section 2.
例题：解 2x² - 3x + 1 = 0。此处 a = 2，b = -3，c = 1，所以 b² - 4ac = 9 - 8 = 1。于是 x = (3 ± 1) / 4，得到 x = 1 或 x = ½，与第 2 节的因式分解结果一致。
stant: solve x² - 4x - 12 = 0. Here a = 1, b = -4, c = -12, so b² - 4ac = 16 + 48 = 64 and √64 = 8. Then x = (4 ± 8) / 2, giving x = 6 or x = -2.
含负常数项的例题：解 x² - 4x - 12 = 0。此处 a = 1，b = -4，c = -12，所以 b² - 4ac = 16 + 48 = 64，√64 = 8。于是 x = (4 ± 8) / 2，得到 x = 6 或 x = -2。
proportion of lost marks in this topic come from writing -4 instead of +4, or from squaring (-4) as -16 instead of +16.
注意当 b 为负时括号的重要性：-(-4) = +4。本主题中失分很大一部分来自把 +4 写成 -4，或者把 (-4)² 算成 -16 而不是 +16。
When the answer is not exact, the question will normally say 'give your answers correct to 3 significant figures' or 'correct to 2 decimal places'. Show the substitution line so that the examiner can award method marks even if your final arithmetic slips.
当答案不是精确值时，题目通常会说明 "答案保留 3 位有效数字" 或 "保留 2 位小数"。请写出代入过程，这样即使最后计算出错，考官仍可给出方法分。
Never round in the middle of a calculation. Keep full accuracy in your calculator and round only at the final step, otherwise your last digit may be wrong.
绝不要在计算中途四舍五入。请在计算器中保留完整精度，只在最后一步取近似值，否则最后一位数字可能出错。
5. The Discriminant | 判别式
The expression b² - 4ac is called the discriminant and is often written as Δ. It tells you how many real roots a quadratic equation has without actually solving it.
表达式 b² - 4ac 称为判别式，通常记作 Δ。它可以在不解方程的情况下告诉你该二次方程有多少个实根。
| Discriminant 判别式 | Number of real roots 实根个数 | Graph 图像 |
| b² - 4ac > 0 | Two distinct real roots 两个不同实根 | Crosses the x-axis twice 与 x 轴相交两次 |
| b² - 4ac = 0 | One repeated root 一个重根 | Touches the x-axis 与 x 轴相切 |
| b² - 4ac < 0 | No real roots 无实根 | Does not meet the x-axis 与 x 轴不相交 |
nd the parabola never touches the x-axis. The completed square form (x + 2)² + 1 confirms this, since the smallest possible value is 1.
例如，对于 x² + 4x + 5 = 0，Δ = 16 - 20 = -4 < 0，所以没有实根，抛物线不与 x 轴相交。配方形式 (x + 2)² + 1 也证实了这一点，因为其最小可能值为 1。
n has equal roots. Set Δ = 0: k² - 36 = 0, so k = 6 or k = -6. Do not forget the negative solution.
扩展课程中非常常见的题型是给出含未知常数的二次方程，例如 x² + kx + 9 = 0，并要求使方程有相等实根的 k 值。令 Δ = 0：k² - 36 = 0，所以 k = 6 或 k = -6。不要忘记负的解。
If a question asks for two distinct real roots, you need Δ > 0, which usually leads to a quadratic inequality in the unknown constant. Solve it using the method in Section 9.
如果题目要求有两个不同实根，则需要 Δ > 0，这通常会得到一个关于未知常数的二次不等式。请用第 9 节的方法求解。
6. Graphs of Quadratic Functions | 二次函数图像
. When a > 0 the parabola opens upwards and has a minimum point; when a < 0 it opens downwards and has a maximum point. The larger the value of |a|, the narrower the parabola.
y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有最低点；当 a < 0 时，开口向下，有最高点。|a| 的值越大，抛物线越窄。
e point to plot first. The x-intercepts, if they exist, are the roots of ax² + bx + c = 0.
令 x = 0 可得 y 轴截距，即 y = c。这是一个快速可靠的首个描点。如果存在 x 轴截距，它们就是方程 ax² + bx + c = 0 的根。
(2a). This line passes through the vertex and lies exactly halfway between the two roots.
每条抛物线都关于一条称为对称轴的竖直线对称。其方程为 x = -b / (2a)。这条线穿过顶点，并且恰好位于两个根的正中间。
axis of symmetry: x = -b / (2a)
at x = (6 + (-2)) / 2 = 2. Substituting back gives the y-coordinate of the turning point.
由于对称性，顶点的横坐标就是两个根的平均值。如果根是 x = 6 和 x = -2，则顶点横坐标为 x = (6 + (-2)) / 2 = 2。代回原式即可得到转折点的纵坐标。
raph of y = x² translated p units to the right and q units up, so its vertex is at (p, q).
在扩展课程中，你可能还会遇到图像变换。y = (x - p)² + q 的图像是 y = x² 向右平移 p 个单位、向上平移 q 个单位得到的，因此其顶点为 (p, q)。
7. Key Features of a Parabola | 抛物线的关键特征
Exam questions on graphs usually ask for specific features rather than a perfect drawing. Being able to extract these features quickly is worth more marks than neat shading.
关于图像的考题通常要求的是具体特征，而不是完美的作图。能快速提取这些特征，比画得好看更能得分。
| Feature 特征 | How to find it 求法 |
| y-intercept y 轴截距 | Set x = 0, y = c 令 x = 0，得 y = c |
| Roots 根 / x-intercepts | Solve ax² + bx + c = 0 解方程 |
| Axis of symmetry 对称轴 | x = -b / (2a) |
| Vertex 顶点 | Complete the square, or average the roots 配方或取两根平均值 |
| Minimum or maximum value 最值 | y-coordinate of the vertex 顶点的纵坐标 |
and the vertex is (2, -16). These four pieces of information are enough for a full sketch.
对于 y = x² - 4x - 12，y 轴截距为 -12，根为 x = 6 和 x = -2。对称轴为 x = 2，顶点为 (2, -16)。这四条信息足以画出完整草图。
When a question asks for the minimum value of a quadratic expression, it is asking for the y-coordinate of the vertex, not the x-coordinate. Read the wording carefully: 'minimum point' means coordinates, 'minimum value' means a single number.
当题目要求二次表达式的最小值时，它问的是顶点的纵坐标，而不是横坐标。请仔细阅读措辞："最低点" 指坐标，"最小值" 指一个数。
8. Sketching and Interpreting Graphs | 图像绘制与识图
A good sketch of a parabola needs four things: the correct orientation, the y-intercept, the x-intercepts where they exist, and the turning point. Label each of these clearly on your diagram.
一条抛物线的合格草图需要四点：正确的开口方向、y 轴截距、存在的 x 轴截距，以及转折点。请在图中清楚地标出每一项。
e for the roots if they are easy to find. Fourth, complete the square to locate the vertex. Finally, join the points with a smooth curve and use symmetry to keep it balanced.
分步方法：第一步，检查 a 的符号以确定开口向上还是向下。第二步，令 x = 0 求 y 轴截距。第三步，如果根容易求就求出根。第四步，配方确定顶点位置。最后，用光滑曲线连接各点，并利用对称性保持图形均衡。
f where the curve crosses the x-axis and give the values to one decimal place if required.
图像
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