📚 Quadratic Equations: Factorisation, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式
Quadratic equations appear throughout the IGCSE Mathematics syllabus, and they are tested in both the Core and Extended papers. This revision guide explains how to recognise a quadratic equation, how to solve it using factorisation, completing the square and the quadratic formula, and how to interpret the discriminant and the graph of a quadratic function. It also includes worked examples and exam tips so you can avoid common errors.
二次方程贯穿 IGCSE 数学课程，无论是 Core 还是 Extended 试卷都有可能考查。本复习指南将讲解如何识别二次方程，如何使用因式分解法、配方法和求根公式求解，以及如何解释判别式和二次函数图像。文中还包含例题与考试技巧，帮助你避免常见错误。
1. What is a Quadratic Equation? | 什么是二次方程？
e written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程是最高次数为 2 的多项式方程，即变量 x 的最高次幂为 2。它总可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。
h is a linear equation, not a quadratic equation. The terms a, b and c are called the quadratic coefficient, linear coefficient and constant term respectively.
条件 a ≠ 0 非常重要。若 a = 0，方程变为 bx + c = 0，这是一次方程而不是二次方程。a、b、c 分别称为二次项系数、一次项系数和常数项。
= x² + 5 is not quadratic because the x² terms cancel out, leaving only a linear equation.
例如，3x² − 7x + 2 = 0 是一个二次方程，其中 a = 3，b = −7，c = 2。但 x² + 2 = x² + 5 不是二次方程，因为 x² 项相互抵消，只剩一次方程。
2. Standard Form and Key Terms | 标准形式与关键术语
This means collecting all terms on one side of the equation, with zero on the other side.
在解二次方程之前，你应始终先将其整理成标准形式 ax² + bx + c = 0。也就是说，将所有项移到方程的一边，另一边为 0。
The solutions of a quadratic equation are also called roots or x-intercepts. They are the values of x that make the equation true. A quadratic equation can have two real roots, one repeated root, or no real roots.
二次方程的解也称为根或 x 轴截距。它们是使方程成立的值。二次方程可以有两个实根、一个重根，或没有实根。
to 2x² − 3x − 6 = 0. Now the equation is ready for factorisation or the quadratic formula.
例题：将 2x² − 5 = 3x + 1 整理成标准形式。把所有项移到左边：2x² − 5 − 3x − 1 = 0，化简得 2x² − 3x − 6 = 0。现在方程就可以使用因式分解或求根公式了。
3. Solving by Factorisation | 因式分解法
r factors. The key principle is the zero product property: if AB = 0, then A = 0 or B = 0.
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。其关键原理是零因子性质：如果 AB = 0，那么 A = 0 或 B = 0。
To solve by factorisation, first set the equation to zero. Then factorise the quadratic expression into two brackets. Finally, set each bracket equal to zero and solve for x.
使用因式分解法时，首先令方程等于 0。然后把二次式分解成两个括号相乘。最后令每个括号分别等于 0，解出 x。
)(x − 3) = 0. Setting each factor to zero gives x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
例题：解 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0。令每个因式等于 0，得 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。
solve 2x² + 7x + 3 = 0. This factorises to (2x + 1)(x + 3) = 0, giving x = −1/2 or x = −3.
当二次项系数不是 1 时，因式分解稍难一些。例如，解 2x² + 7x + 3 = 0。它可以分解为 (2x + 1)(x + 3) = 0，得到 x = −1/2 或 x = −3。
Always check your factorisation by expanding the brackets. A common mistake is to choose factors that give the correct constant term but the wrong middle coefficient.
始终通过展开括号来检验因式分解是否正确。常见错误是所选的因式常数项正确，但中间一次项系数不对。
4. Solving by Completing the Square | 配方法
l when factorisation is difficult or when you need to find the turning point of the graph.
配方法将二次方程改写成 (x + p)² = q 的形式。当因式分解较难，或需要求图像顶点时，配方法特别有用。
steps are: move the constant term to the other side, add (b/2)² to both sides, write the left side as a perfect square, then take square roots.
对于首项系数为 1 的二次方程 x² + bx + c = 0，步骤如下：把常数项移到另一边，两边同时加上 (b/2)²，左边写成完全平方，然后开平方。
rite as a square: (x + 3)² = 4. Taking square roots gives x + 3 = ±2, so x = −1 or x = −5.
例题：解 x² + 6x + 5 = 0。移常数项：x² + 6x = −5。两边加 (6/2)² = 9：x² + 6x + 9 = 4。写成完全平方：(x + 3)² = 4。开平方得 x + 3 = ±2，所以 x = −1 或 x = −5。
² + 4x + 3 = 0 before completing the square. This gives (x + 2)² = 1, so x = −1 or x = −3.
如果系数 a 不等于 1，先让每一项除以 a。例如，2x² + 8x + 6 = 0 先化为 x² + 4x + 3 = 0，再配方。得到 (x + 2)² = 1，所以 x = −1 或 x = −3。
5. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation in standard form. It is derived from completing the square and is given by the equation below.
求根公式可以解任何标准形式的二次方程。它由配方法推导而来，公式如下。
x = (−b ± √b² − 4ac) / 2a
refully, especially when b or c is negative. Use brackets around negative numbers when squaring.
其中 a、b、c 是 ax² + bx + c = 0 中的系数。代入时务必仔细，尤其是 b 或 c 为负数时。平方负数时要使用括号。
−1. The discriminant is b² − 4ac = (−4)² − 4 × 2 × (−1) = 16 + 8 = 24. Therefore x = (4 ± √24) / 4 = (4 ± 2√6) / 4 = 1 ± √6/2.
例题：用公式解 2x² − 4x − 1 = 0。这里 a = 2，b = −4，c = −1。判别式为 b² − 4ac = (−4)² − 4 × 2 × (−1) = 16 + 8 = 24。因此 x = (4 ± √24) / 4 = (4 ± 2√6) / 4 = 1 ± √6/2。
The formula is especially useful when the quadratic expression cannot factorise neatly. In an exam, you may be asked to give answers in surd form or to a specified degree of accuracy.
当二次式不能方便地因式分解时，求根公式尤其有用。考试中可能要求以根式形式给出答案，或精确到指定小数位。
6. Discriminant and Nature of Roots | 判别式与根的性质
c. It tells you how many real roots a quadratic equation has without solving the equation.
判别式是表达式 D = b² − 4ac。它不需要解方程就能告诉你二次方程有多少个实根。
it has one repeated real root. If D < 0, it has no real roots, only two complex roots.
如果 D > 0，方程有两个不同的实根。如果 D = 0，方程有一个重根。如果 D < 0，方程没有实根，只有两个复数根。
s x = 2. The equation x² + x + 1 = 0 has discriminant 1 − 4 = −3, so it has no real roots.
例如，x² − 4x + 4 = 0 的判别式为 (−4)² − 4 × 1 × 4 = 0，因此它有一个重根 x = 2。方程 x² + x + 1 = 0 的判别式为 1 − 4 = −3，因此没有实根。
Some IGCSE questions ask you to find the range of k for which a quadratic equation has two distinct real roots. You set up the inequality D > 0 and solve for k.
有些 IGCSE 题目要求你求 k 的取值范围，使二次方程有两个不同实根。你需要建立不等式 D > 0，并解出 k。
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.
函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，有最小值点。如果 a < 0，抛物线开口向下，有最大值点。
und by substituting this x-value back into the equation. The roots of the equation are the points where the graph crosses the x-axis.
顶点坐标的 x 值为 x = −b / 2a。将这个 x 值代回原方程即可求出顶点的 y 坐标。方程的根就是图像与 x 轴交点的横坐标。
If the discriminant is positive, the parabola crosses the x-axis at two points. If it is zero, the parabola touches the x-axis at one point. If it is negative, the parabola does not cross the x-axis at all.
如果判别式为正，抛物线与 x 轴相交于两点。如果判别式为零，抛物线与 x 轴相切于一点。如果判别式为负，抛物线完全不与 x 轴相交。
You should be able to sketch a quadratic graph by finding the y-intercept, the roots and the vertex. Label these clearly on your graph in an exam.
你应当能够通过求出 y 轴截距、根和顶点来画出二次函数图像的草图。考试时请在图上清楚标注这些点。
8. Applications and Word Problems | 应用与文字题
Quadratic equations often appear in word problems involving area, projectiles, profit or consecutive integers. The key is to translate the written information into a quadratic equation, then solve and check that your answer makes sense in context.
二次方程经常出现在涉及面积、抛射体、利润或连续整数的文字题中。关键是把文字信息转化为二次方程，然后求解，并检查答案在实际情况中是否合理。
(x − 8) = 0, so x = 8 or x = −9. Since the integers are positive, the numbers are 8 and 9.
例题：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，得到 x² + x − 72 = 0。因式分解得 (x + 9)(x − 8) = 0，所以 x = 8 或 x = −9。由于整数为正，答案是 8 和 9。
0 = 0, so w = 5 or w = −8. The width must be positive, so w = 5 cm and the length is 8 cm.
另一种常见题型是矩形问题。如果矩形的长比宽多 3 cm，面积为 40 cm²，设宽为 w。则 w(w + 3) = 40，得 w² + 3w − 40 = 0，所以 w = 5 或 w = −8。宽必须为正，所以 w = 5 cm，长为 8 cm。
Always reject solutions that are not possible in the real situation, such as negative lengths or negative time. This is an important step in IGCSE word problems.
务必舍去在实际情况中不可能的解，例如负的长度或负的时间。这在 IGCSE 文字题中是重要的一步。
9. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is forgetting to set the equation to zero before factorising. You cannot factorise correctly if the equation is not in standard form.
一个常见错误是在因式分解前忘记令方程等于 0。如果方程不是标准形式，就无法正确因式分解。
Another error is mishandling signs in the quadratic formula. Always write b and c with their signs, and use brackets when substituting negative values into b² − 4ac.
另一个错误是在求根公式中处理符号不当。务必带上 b 和 c 的符号，并在代入负值计算 b² − 4ac 时使用括号。
Some students factorise incorrectly and forget to check by expansion. Many exam marks are lost on simple arithmetic mistakes. Always check your roots by substituting them back into the original equation.
一些学生因式分解错误，而且忘记通过展开来检验。许多考试失分来自简单的算术错误。请始终将求得的根代入原方程进行检验。
When using completing the square, remember that (x + p)² = q leads to x = −p ± √q, not x = p ± √q. The sign of p inside the bracket is reversed.
使用配方法时，记住 (x + p)² = q 导出 x = −p ± √q，而不是 x = p ± √q。括号内 p 的符号要变号。
- English: Rearrange the equation to standard form ax² + bx + c = 0 before solving. 中文：解题前先将方程化为标准形式 ax² + bx + c = 0。
- English: Choose the most efficient method: factorisation first, then formula or completing the square. 中文：选择最高效的方法：先尝试因式分解，再用公式或配方法。
- English: Show every step clearly to gain method marks even if the final answer is wrong. 中文：清晰展示每一步，即使最终答案错误也能获得方法分。
- English: Write exact answers as surds unless the question asks for a decimal approximation. 中文：除非题目要求小数近似值，否则答案保留根式形式。
10. Summary and Practice Checklist | 总结与练习清单
To master quadratic equations for IGCSE Mathematics, you need to be confident with all three algebraic methods: factorisation, completing the square and the quadratic formula. You must also understand the discriminant and the graph of a quadratic function.
要在 IGCSE 数学中掌握二次方程，你需要熟练运用三种代数方法：因式分解法、配方法和求根公式。你还需要理解判别式和二次函数图像。
Use the checklist below to guide your revision. Tick each item when you can complete it without help, and work through IGCSE past paper questions to build speed and
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