📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | IGCSE数学二次方程：因式分解、配方法与求根公式
Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. They appear in algebra, graphing, and real-world problems such as area, projectile motion, and profit calculations. In this article, we will work through the three main solution methods - factorising, completing the square, and the quadratic formula - and connect them to the graph of a quadratic function.
二次方程是 IGCSE 数学大纲中最重要的专题之一。它们出现在代数、函数图像以及面积、抛体运动和利润计算等实际问题中。本文将逐一讲解三种主要解法——因式分解法、配方法和求根公式法——并将它们与二次函数的图像联系起来。
1. What is a Quadratic Equation? | 什么是二次方程？
d a ≠ 0. The condition a ≠ 0 is essential because if a were equal to zero, the x² term would disappear and the equation would become linear, not quadratic.
二次方程是任何可以写成 a x² + b x + c = 0 形式的方程，其中 a、b 和 c 是常数，并且 a ≠ 0。a ≠ 0 这个条件非常重要，因为如果 a 等于 0，x² 项就会消失，方程就变成一次方程，而不是二次方程。
a x² + b x + c = 0, a ≠ 0
12 or (x − 2)(x + 3) = 0 are also quadratic once they are expanded or rearranged. The word "quadratic" comes from the Latin word for "square", which reminds us that the key feature is the squared term.
例如，x² − 5x + 6 = 0 是二次方程，因为 x 的最高次数是 2。像 3x² = 12 或 (x − 2)(x + 3) = 0 这样的方程在展开或移项后也都是二次方程。"二次"一词来源于拉丁语中表示"平方"的词，这提醒我们关键特征是平方项。
A quadratic equation can have two real roots, one repeated real root, or no real roots. These three cases are controlled by the discriminant, which we will study later. In IGCSE exams, you are usually expected to solve quadratic equations exactly, often leaving surds in simplified form unless the question asks for a decimal answer.
二次方程可以有两个实根、一个重根，或者没有实根。这三种情况由判别式控制，我们稍后会学习。在 IGCSE 考试中，通常要求精确求解二次方程，除非题目要求小数答案，否则结果常以最简根式形式保留。
2. Standard Form and Key Terms | 标准形式与关键术语
The value of a cannot be zero, but b and c may be zero. For example, x² − 9 = 0 has b = 0, and 2x² + 5x = 0 has c = 0.
二次方程的标准形式是 a x² + b x + c = 0。在这个形式中，a 是 x² 的系数，b 是 x 的系数，c 是常数项。a 的值不能为零，但 b 和 c 可以为零。例如，x² − 9 = 0 中 b = 0，2x² + 5x = 0 中 c = 0。
. For instance, if you are given 3x² = 12, rewrite it as 3x² − 12 = 0. If you are given (x − 2)(x + 3) = 0, it is already in factorised form, but if it were (x − 2)(x + 3) = 6, you would need to expand and rearrange first.
在使用任何解法之前，一定要把方程整理成一边恰好为零。这是最常见的失分点之一。例如，如果题目给出 3x² = 12，应改写为 3x² − 12 = 0。如果题目给出 (x − 2)(x + 3) = 0，它已经是因式分解形式；但如果是 (x − 2)(x + 3) = 6，则需要先展开并移项。
| Term | Meaning | 中文 |
|---|---|---|
| a | coefficient of x², a ≠ 0 | x² 的系数，a ≠ 0 |
| b | coefficient of x | x 的系数 |
| c | constant term | 常数项 |
| root | a value of x that makes the equation true | 使方程成立的 x 值 |
ue gives zero. In graph terms, roots are the x-intercepts of the curve y = a x² + b x + c.
根也称为零点或解。例如，x = 2 和 x = 3 是 x² − 5x + 6 = 0 的根，因为代入任何一个值都得到零。从图像上看，根就是曲线 y = a x² + b x + c 与 x 轴交点的横坐标。
3. Solving by Factorising | 因式分解法
roduct of two factors is zero, then at least one of the factors must be zero. In symbols, if A × B = 0, then A = 0 or B = 0.
当二次式能整齐地分解为整数因式时，因式分解法通常是最快的方法。该方法依据零乘积法则：如果两个因式的乘积为零，那么至少有一个因式为零。用符号表示，如果 A × B = 0，那么 A = 0 或 B = 0。
need two numbers that multiply to 6 and add to −5. These are −2 and −3, so the equation becomes (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.
对于首项系数为 1 的简单二次方程 x² + b x + c = 0，要找两个数，它们的乘积为 c，和为 b。例如，解 x² − 5x + 6 = 0 时，需要两个数乘积为 6、和为 −5，这两个数是 −2 和 −3，因此方程变为 (x − 2)(x − 3) = 0。所以 x = 2 或 x = 3。
2 × 3 = 6, and we need two numbers that multiply to 6 and add to 7. These are 1 and 6. Split the middle term: 2x² + x + 6x + 3 = 0, then factor in pairs: x(2x + 1) + 3(2x + 1) = 0, so (2x + 1)(x + 3) = 0. Hence x = −1/2 or x = −3.
当首项系数不为 1 时，先计算 a × c，再找两个数，使它们的乘积为 a × c、和为 b。例如，在 2x² + 7x + 3 = 0 中，a × c = 2 × 3 = 6，需要两个数乘积为 6、和为 7，这两个数是 1 和 6。拆分中间项：2x² + x + 6x + 3 = 0，然后分组分解：x(2x + 1) + 3(2x + 1) = 0，所以 (2x + 1)(x + 3) = 0。因此 x = −1/2 或 x = −3。
Difference of two squares: x² − 16 = 0 becomes (x − 4)(x + 4) = 0, so x = 4 or x = −4.
平方差：x² − 16 = 0 变为 (x − 4)(x + 4) = 0，所以 x = 4 或 x = −4。
Common factor: 2x² + 5x = 0 becomes x(2x + 5) = 0, so x = 0 or x = −5/2.
提取公因式：2x² + 5x = 0 变为 x(2x + 5) = 0，所以 x = 0 或 x = −5/2。
Perfect square: x² + 10x + 25 = 0 becomes (x + 5)² = 0, so x = −5 is a repeated root.
完全平方：x² + 10x + 25 = 0 变为 (x + 5)² = 0，所以 x = −5 是重根。
Always check your factorisation by expanding it back. If the expanded form does not match the original quadratic, correct the signs or the numbers before solving. Also remember that not every quadratic factorises with integers; in that case, use completing the square or the quadratic formula.
一定要把因式分解结果展开回去检验。如果展开后与原二次式不符，先修正符号或数字再求解。还要记住，并非所有二次式都能用整数因式分解；这时应使用配方法或求根公式。
4. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic expression as a perfect square plus or minus a constant. It works for every quadratic, even when factorising fails, and it is also the key to finding the vertex of a parabola. The basic idea for x² + b x + c is to write it as (x + b/2)² − (b/2)² + c.
配方法把二次式改写为一个完全平方加减一个常数。它对所有二次式都适用，即使因式分解失败也能用；同时它也是求抛物线顶点的关键。对于 x² + b x + c，基本思路是把它写成 (x + b/2)² − (b/2)² + c。
x² + b x + c = (x + b/2)² − (b/2)² + c
equation becomes (x + 3)² − 9 + 1 = 0, so (x + 3)² − 8 = 0. Hence (x + 3)² = 8, x + 3 = ±√8, and x = −3 ± 2√2. These are the exact roots.
例：用配方法解 x² + 6x + 1 = 0。6 的一半是 3，所以把 x² + 6x 写成 (x + 3)² − 9。方程变为 (x + 3)² − 9 + 1 = 0，即 (x + 3)² − 8 = 0。因此 (x + 3)² = 8，x + 3 = ±√8，x = −3 ± 2√2。这就是精确根。
− 2)² − 3. To solve 2x² − 8x + 5 = 0, set 2(x − 2)² − 3 = 0, so (x − 2)² = 3/2, x − 2 = ±√3/2, and x = 2 ± √6/2 after simplifying the surd.
当 a ≠ 1 时，先把 a 从 x² 和 x 项中提取出来，不要从常数项中提取。例如，2x² − 8x + 5 = 2(x² − 4x) + 5 = 2[(x − 2)² − 4] + 5 = 2(x − 2)² − 8 + 5 = 2(x − 2)² − 3。解 2x² − 8x + 5 = 0 时，令 2(x − 2)² − 3 = 0，则 (x − 2)² = 3/2，x − 2 = ±√3/2，化简根式后得 x = 2 ± √6/2。
Completing the square is especially useful when the question asks for exact surd answers or when it asks you to show that a quadratic can be written in a certain form. It is also the method behind the quadratic formula, which we derive next.
当题目要求精确根式答案，或要求证明二次式可写成某种形式时，配方法尤其有用。它也是求根公式背后的方法，我们接下来就推导它。
5. Deriving and Using the Quadratic Formula | 推导与使用求根公式
tarting with a x² + b x + c = 0, divide by a: x² + (b/a)x + c/a = 0. Move the constant: x² + (b/a)x = −c/a. Add (b/2a)² to both sides to complete the square.
求根公式可以解任何标准形式的二次方程。它通过对一般式 a x² + b x + c = 0 配方推导得到。从 a x² + b x + c = 0 开始，两边除以 a：x² + (b/a)x + c/a = 0。移常数项：x² + (b/a)x = −c/a。两边加上 (b/2a)² 来完成配方。
x = (−b ± √b² − 4ac) / (2a)
ed square form is (x + b/2a)² = (b² − 4ac) / (4a²). Taking square roots gives x + b/2a = ±√b² − 4ac e. The ± sign is essential because a square root has both a positive and a negative value.
配方后的形式是 (x + b/2a)² = (b² − 4ac) / (4a²)。开平方得 x + b/2a = ±√b² − 4ac / (2a)，再减去 b/2a 就得到上面的公式。± 号必不可少，因为平方根有正负两个值。
4, c = −3. Substitute into the formula: x = (4 ± √((−4)² − 4 × 2 × (−3))) / (2 × 2) = (4 ± √16 + 24) / 4 = (4 ± √40) / 4. Since √40 = 2√10, we get x = (4 ± 2√10) / 4 = 1 ± √10/2.
例：解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入公式：x = (4 ± √((−4)² − 4 × 2 × (−3))) / (2 × 2) = (4 ± √16 + 24) / 4 = (4 ± √40) / 4。因为 √40 = 2√10，所以 x = (4 ± 2√10) / 4 = 1 ± √10/2。
Common mistakes with the formula include forgetting the negative sign in −b, using b instead of b² inside the square root, and dividing only part of the numerator by 2a. Write down a, b and c clearly before substituting, and use brackets when entering negative numbers into a calculator.
使用公式时的常见错误包括：忘记 −b 中的负号、在根号内用了 b 而不是 b²、以及只把分子的一部分除以 2a。代入前先写清楚 a、b、c，并在计算器中输入负数时使用括号。
6. The Discriminant and the Nature of Roots | 判别式与根的性质
The expression inside the square root, b² − 4ac, is called the discriminant. It is often denoted by Δ. The discriminant tells you how many real roots a quadratic equation has without actually solving it.
根号内的表达式 b² − 4ac 称为判别式，通常用 Δ 表示。判别式可以在不解方程的情况下告诉你二次方程有多少个实根。
Δ = b² − 4ac
| Discriminant | Nature of roots | 中文说明 |
|---|---|---|
| Δ > 0 | two distinct real roots | 两个不同的实根 |
| Δ = 0 | one repeated real root | 一个重实根 |
| Δ < 0 | no real roots | 没有实根 |
d root, x = 3. For x² − 5x + 6 = 0, Δ = 25 − 24 = 1 > 0, so there are two distinct real roots, x = 2 and x = 3.
例如，对于 x² + 2x + 5 = 0，Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16。因为 Δ < 0，所以方程没有实根。对于 x² − 6x + 9 = 0，Δ = 36 − 36 = 0，所以有一个重根 x = 3。对于 x² − 5x + 6 = 0，Δ = 25 − 24 = 1 > 0，所以有两个不同实根 x = 2 和 x = 3。
at one point; if Δ < 0, it does not cross the x-axis at all. This connection is frequently tested in IGCSE examination questions.
从图像上看，判别式告诉你抛物线与 x 轴相交多少次。如果 Δ > 0，相交两次；如果 Δ = 0，只在一点与 x 轴相切；如果 Δ < 0，则完全不相交。这一联系在 IGCSE 考试中经常考查。
7. The Graph of a Quadratic Function | 二次函数的图像
. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The larger the value of |a|, the narrower the parabola; the smaller the value of |a|, the wider it becomes.
y = a x² + b x + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，有最低点；如果 a < 0，开口向下，有最高点。|a| 越大，抛物线越窄；|a| 越小，抛物线越宽。
ry is the vertical line x = −b/(2a), and the vertex lies on this line. The vertex coordinates can be found by completing the square or by using x = −b/(2a) and then substituting to find y.
方程 a x² + b x + c = 0 的根就是图像与 x 轴交点的横坐标。y 轴截距是 c，因为代入 x = 0 得 y = c。对称轴是竖直线 x = −b/(2a)，顶点就在这条线上。顶点坐标可以通过配
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图