📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握IGCSE数学二次方程
Quadratic equations are at the heart of IGCSE Mathematics. They appear in algebra, graphs, number problems and even geometry. This revision guide explains how to recognise a quadratic, how to solve it by factorising, completing the square or using the quadratic formula, and how to interpret its discriminant and graph. Work through each section and you will be ready for both Core and Extended papers.
二次方程是IGCSE数学的核心内容，出现在代数、图像、数字问题甚至几何中。本复习指南将讲解如何识别二次方程，如何用因式分解法、配方法和求根公式求解，以及如何理解判别式和图像。逐一学习每个小节后，你将能从容应对Core和Extended试卷。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be written in the standard form shown below, where a, b and c are constants and a ≠ 0. The key feature is the x² term. If the highest power of x is 2, the equation is quadratic.
二次方程是任何可以写成下面标准形式的方程，其中a、b、c为常数且a ≠ 0。关键特征是x²项。如果x的最高次数是2，该方程就是二次方程。
ax² + bx + c = 0
You will often see equations such as x² − 5x + 6 = 0, 2x² + 3x − 2 = 0, or x² − 9 = 0. Some need rearranging before you can identify a, b and c.
你经常会看到诸如 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 或 x² − 9 = 0 的方程。有些方程需要先移项才能识别出a、b和c。
2. Standard Form and Key Terms | 标准形式与关键术语
For the equation ax² + bx + c = 0, the constant a is called the leading coefficient, b is the linear coefficient, and c is the constant term. Keep the equation equal to 0 before applying most solution methods.
对于方程 ax² + bx + c = 0，常数a称为二次项系数，b称为一次项系数，c称为常数项。在应用大多数解法之前，请先让方程右边等于0。
Example: rewrite 3x² = 7 − 2x as 3x² + 2x − 7 = 0. Here a = 3, b = 2 and c = −7.
例如：将 3x² = 7 − 2x 改写为 3x² + 2x − 7 = 0。此时 a = 3，b = 2，c = −7。
Always collect all terms on one side and arrange them in descending powers of x. This makes factorising and substitution much easier.
请始终把所有项移到同一边，并按x的降幂排列。这样做会让因式分解和代入公式容易得多。
3. Solving by Factorising | 因式分解法
Factorising works when the quadratic expression can be written as a product of two linear brackets. Start by finding two numbers that multiply to give ac and add to give b, then split the middle term or inspect directly.
当二次式可以写成两个一次括号的乘积时，因式分解法非常有效。首先找到两个数，它们的乘积为ac，和为b，然后拆分中间项或直接观察。
x² − 5x + 6 = (x − 2)(x − 3)
For x² − 5x + 6, the numbers −2 and −3 multiply to give +6 and add to give −5, so the factorised form is (x − 2)(x − 3).
对于 x² − 5x + 6，数 −2 和 −3 相乘得 +6，相加得 −5，因此因式分解为 (x − 2)(x − 3)。
When a is not 1, such as in 2x² + 3x − 2, look for two numbers that multiply to give 2 × (−2) = −4 and add to give 3. These are 4 and −1, so the expression can be written as 2x² + 4x − x − 2 = (2x − 1)(x + 2).
当a不等于1时，例如 2x² + 3x − 2，要找两个数，它们的乘积为 2 × (−2) = −4，和为3。这两个数是4和−1，因此该式可写成 2x² + 4x − x − 2 = (2x − 1)(x + 2)。
4. The Zero Product Property | 零乘积性质
The zero product property states that if (x − p)(x − q) = 0, then x − p = 0 or x − q = 0. Therefore x = p or x = q.
零乘积性质指出，如果 (x − p)(x − q) = 0，则 x − p = 0 或 x − q = 0，因此 x = p 或 x = q。
(x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3
Never divide both sides by a bracket that contains x, because you might lose a solution. This is a very common mistake in IGCSE algebra questions.
切勿两边同时除以含有x的括号，因为你可能会丢失一个解。这是IGCSE代数题中非常常见的错误。
5. Solving by Completing the Square | 配方法
To solve by completing the square, first make the coefficient of x² equal to 1. Then write x² + bx as (x + b/2)² − (b/2)² and keep the equation balanced.
用配方法求解时，首先使x²的系数等于1。然后将 x² + bx 写成 (x + b/2)² − (b/2)²，并保持等式平衡。
x² + 6x + 2 = 0 ⇒ (x + 3)² − 9 + 2 = 0 ⇒ (x + 3)² = 7
Then take the square root of both sides and remember the ± sign: x + 3 = ±√7, so x = −3 ± √7.
然后两边开平方，并记住正负号：x + 3 = ±√7，所以 x = −3 ± √7。
If a is not 1, divide the whole equation by a first. Completing the square is also the method used to find the turning point of a quadratic graph.
如果a不等于1，请先让整个方程除以a。配方法也是求二次图像转折点所用的方法。
6. The Quadratic Formula | 求根公式
The quadratic formula gives the solutions of ax² + bx + c = 0 directly. It is provided on some IGCSE formula sheets, but you must know how to substitute correctly.
求根公式可以直接给出 ax² + bx + c = 0 的解。某些IGCSE公式表会提供该公式，但你必须知道如何正确代入。
x = (−b ± √(b² − 4ac)) / (2a)
Substitute a, b and c carefully, especially when they are negative. This method works for all quadratics, even when factorising is difficult or impossible.
代入a、b和c时要格外小心，尤其是它们为负数时。该方法适用于所有二次方程，即使因式分解很困难或无法进行。
7. The Discriminant | 判别式
The discriminant is the part under the square root: Δ = b² − 4ac. It tells you how many real solutions the quadratic equation has.
判别式是平方根号下的部分：Δ = b² − 4ac。它告诉你二次方程有多少个实数解。
If Δ > 0, there are two distinct real solutions. If Δ = 0, there is exactly one real solution, called a repeated root. If Δ < 0, there are no real solutions.
如果 Δ > 0，有两个不同的实数解；如果 Δ = 0，只有一个实数解，称为重根；如果 Δ < 0，没有实数解。
You can use the discriminant before solving to check how many answers to expect. Exam questions often ask about the number of roots without requiring the actual roots.
你可以在求解前使用判别式来检验应有多少个答案。考试题经常只问根的个数，而不要求求出具体的根。
8. Graphs of Quadratic Functions | 二次函数图像
The graph of y = ax² + bx + c is a parabola. 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。如果 a > 0，抛物线开口向上且有最低点；如果 a < 0，抛物线开口向下且有最高点。
The x-intercepts of the graph are the real solutions of ax² + bx + c = 0. The y-intercept is (0, c). The vertex or turning point can be found by completing the square or using x = −b / (2a).
图像与x轴的交点就是 ax² + bx + c = 0 的实数解。与y轴的交点为 (0, c)。顶点或转折点可以通过配方法或 x = −b / (2a) 求得。
Sketching the graph requires labelling the turning point, the y-intercept and any x-intercepts clearly. This is a frequent IGCSE Extended topic.
画图像时需要清楚标出转折点、y轴截距以及所有x轴截距。这是IGCSE Extended部分经常考查的内容。
9. Applications and Word Problems | 应用与文字题
Quadratic equations often model area, projectile motion, product puzzles and economics problems. When solving a word problem, define the unknown, form a quadratic, solve it, and then decide whether each solution makes sense in context.
二次方程常用于建模面积、抛体运动、乘积谜题和经济问题。解应用题时，先设未知数，列出二次方程，求解，然后判断每个解在题意中是否合理。
Example: The product of two consecutive positive integers is 72. Let the integers be x and x + 1. Then x(x + 1) = 72, so x² + x − 72 = 0, giving x = 8 or x = −9. Since the integers are positive, x = 8 and the numbers are 8 and 9.
例如：两个连续正整数的乘积为72。设这两个整数为x和x+1。则 x(x + 1) = 72，即 x² + x − 72 = 0，解得 x = 8 或 x = −9。由于要求正整数，所以 x = 8，这两个数是8和9。
Always discard solutions that do not fit the real-world situation, such as negative lengths or times.
请始终舍去不符合现实情境的解，例如负的长度或时间。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common mistakes include forgetting to set the equation to zero, ignoring the negative sign in b or c, losing the ± when taking square roots, and dividing by a bracket containing x. Take your time with signs and always check your answers by substitution.
常见错误包括忘记将方程设为零、忽略b或c中的负号、开平方时丢失正负号、以及除以含有x的括号。请仔细处理符号，并始终用代入法检查答案。
Write down a, b and c before substituting into the formula. Check factorisation by expanding the brackets. Use the discriminant to check the number of solutions. Give exact answers unless the question asks for decimals.
代入公式前先写出a、b和c；通过展开括号检查因式分解；用判别式检验解的个数；除非题目要求小数，否则给出精确答案。
11. Practice Problem Walkthrough | 练习题讲解
Let us solve 2x² − 3x − 5 = 0 using the quadratic formula. Here a = 2, b = −3 and c = −5.
我们使用求根公式解 2x² − 3x − 5 = 0。这里 a = 2，b = −3，c = −5。
x = (−(−3) ± √((−3)² − 4 × 2 × (−5))) / (2 × 2)
This simplifies to x = (3 ± √(9 + 40)) / 4 = (3 ± √49) / 4 = (3 ± 7) / 4, giving x = 10/4 = 5/2 or x =
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