📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与二次公式法
A quadratic equation is one of the most important algebraic skills in the IGCSE Mathematics syllabus. You are expected to solve quadratics fluently using factorising, completing the square, and the quadratic formula, and to interpret the discriminant in relation to the number of real roots.
二次方程是 IGCSE 数学大纲中最重要的代数技能之一。你需要熟练运用因式分解法、配方法和二次公式法来解二次方程，并能通过判别式判断实根的个数。
1. What Is a Quadratic Equation? | 什么是二次方程？
are constants and a ≠ 0. The highest power of the unknown x is 2, which is why it is called quadratic.
二次方程是指形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。未知数 x 的最高次数为 2，因此称为二次方程。
ather than quadratic. In IGCSE questions, the equation may be given in an expanded form, and you need to rearrange it into standard form before solving.
如果 a = 0，方程就变成一次方程而非二次方程。在 IGCSE 题目中，方程可能以展开形式给出，你需要先将其整理成标准形式再求解。
2. Standard Form and Key Terms | 标准形式与关键术语
e quadratic term, bx is the linear term, and c is the constant term. The values of x that satisfy the equation are called roots or solutions.
二次方程的标准形式为 ax² + bx + c = 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。满足方程的 x 值称为根或解。
e and zero on the other. For example, x² + 4 = 5x should be rearranged to x² − 5x + 4 = 0.
在应用任何方法之前，应把所有项移到等号一侧，令另一侧为零。例如 x² + 4 = 5x 应整理为 x² − 5x + 4 = 0。
3. Solving by Factorising | 因式分解法
m x² + bx + c = 0, you need to find two numbers that multiply to give c and add to give b.
当二次方程有简单的整数根时，因式分解法通常是最快的方法。对于形如 x² + bx + c = 0 的二次式，你需要找到两个数，它们的乘积等于 c，和等于 b。
If the coefficient of x² is not 1, you can split the middle term, use trial and error, or factor out a common factor first. Always set each bracket equal to zero and solve for x.
如果 x² 的系数不是 1，可以拆中项、进行试探组合，或先提取公因式。最后一定要令每个括号分别等于零并解出 x。
4. Worked Example: Factorising | 因式分解示例
roduct of 6 and a sum of 5: these are 2 and 3. Therefore x² + 5x + 6 = (x + 2)(x + 3) = 0.
解方程 x² + 5x + 6 = 0。找两个数，乘积为 6、和为 5：这两个数是 2 和 3。因此 x² + 5x + 6 = (x + 2)(x + 3) = 0。
tions are x = −2 and x = −3. Always check by substituting back into the original equation.
令每个因式为零，得 x + 2 = 0 或 x + 3 = 0，所以解为 x = −2 和 x = −3。求出后应代回原方程进行检验。
5. Completing the Square | 配方法
Completing the square is a powerful method that always works, even when the roots are not integers. The idea is to rewrite ax² + bx + c as a perfect square plus or minus a constant.
配方法是一种始终可行的方法，即使根不是整数也能使用。其核心思想是将 ax² + bx + c 改写为一个完全平方加上或减去一个常数。
For x² + bx, take half of b, square it, and add it to both sides. This gives (x + b/2)². For example, x² + 6x becomes (x + 3)² − 9.
对于 x² + bx，取 b 的一半，平方后加到等式两边，即可得到 (x + b/2)²。例如 x² + 6x 可写成 (x + 3)² − 9。
x² + bx = (x + b/2)² − (b/2)²
6. Worked Example: Completing the Square | 配方法示例
right: x² + 6x = −2. Half of 6 is 3, and 3² = 9, so add 9 to both sides: x² + 6x + 9 = 7.
解方程 x² + 6x + 2 = 0。将常数项移到右边：x² + 6x = −2。6 的一半是 3，3² = 9，所以两边同时加 9：x² + 6x + 9 = 7。
he left side is a perfect square: (x + 3)² = 7. Take square roots on both sides: x + 3 = ±√7. Therefore x = −3 ± √7.
此时左边为完全平方：(x + 3)² = 7。两边开平方得 x + 3 = ±√7。因此 x = −3 ± √7。
7. The Quadratic Formula | 二次公式法
derived from completing the square and is especially useful when factorising is difficult.
二次公式法可以直接给出任何二次方程 ax² + bx + c = 0 的解。它由配方法推导而来，在因式分解困难时尤其有用。
x = (−b ± √b² − 4ac) ÷ (2a)
You must substitute a, b and c carefully, including negative signs. Simplify the numerator step by step before dividing by 2a.
代入 a、b、c 时必须注意符号，尤其是负号。分子先逐步化简，再除以 2a。
8. Worked Example: Quadratic Formula | 二次公式法示例
0 using the quadratic formula. Here a = 1, b = −4, and c = −5. Substituting gives x = (4 ± √16 + 20) ÷ 2 = (4 ± √36) ÷ 2 = (4 ± 6) ÷ 2.
用二次公式法解方程 x² − 4x − 5 = 0。这里 a = 1，b = −4，c = −5。代入公式得 x = (4 ± √16 + 20) ÷ 2 = (4 ± √36) ÷ 2 = (4 ± 6) ÷ 2。
(4 + 6) ÷ 2 = 5 and x = (4 − 6) ÷ 2 = −1. The formula gives both roots in one calculation.
两个解为 x = (4 + 6) ÷ 2 = 5 和 x = (4 − 6) ÷ 2 = −1。该公式一次计算就能同时得出两个根。
9. Discriminant and Nature of Roots | 判别式与根的性质
the discriminant. It tells you the nature of the roots without fully solving the equation.
平方根内的表达式 Δ = b² − 4ac 称为判别式。它可以在不完整求解方程的情况下判断根的性质。
- If Δ > 0, there are two distinct real roots.
- 若 Δ > 0，方程有两个不相等的实根。
- If Δ = 0, there is one repeated real root.
- 若 Δ = 0，方程有一个重根。
- If Δ < 0, there are no real roots, but two complex roots.
- 若 Δ < 0，方程无实根，但有两个复数根。
10. Choosing the Best Method | 选择最佳方法
The best method depends on the form of the quadratic. Factorising is fastest when the roots are integers and the coefficient of x² is simple. Completing the square is useful for deriving the vertex form or solving when roots are irrational.
最佳方法取决于二次方程的形式。当根为整数且 x² 的系数简单时，因式分解法最快。配方法适用于求顶点式或根为无理数的情况。
The quadratic formula is the most general method and should be used when factorising is not obvious. In exams, always check whether the question specifies a particular method.
二次公式法是最通用的方法，在因式分解不明显时应优先使用。考试时，一定要看清题目是否指定了某种特定方法。
| Method | Best for | Example clue |
|---|---|---|
| Factorising | Integer roots, a = 1 | x² + 5x + 6 = 0 |
| Completing the square | Vertex form, irrational roots | x² + 6x + 2 = 0 |
| Quadratic formula | Any quadratic | 2x² − 3x − 4 = 0 |
方法 | 最适合的情形 | 示例线索。因式分解法适合整数根且 a = 1 的情形，例如 x² + 5x + 6 = 0；配方法适合求顶点式或无理根，例如 x² + 6x + 2 = 0；二次公式法适用于任何二次方程，例如 2x² − 3x − 4 = 0。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common mistake is forgetting to set the equation to zero before factorising. Another is mishandling negative signs when using the quadratic formula, especially when b or c is negative.
一个常见错误是在因式分解前忘记将方程整理为零的形式。另一个常见错误是在使用二次公式时处理负号不当，尤其是 b 或 c 为负数时。
Always check your roots by substitution, and write solutions in the required form, such as exact surds or rounded decimals. Show all steps clearly because method marks are awarded in IGCSE exams.
一定要通过代入验算根是否正确，并按要求的形式写出解，例如保留根号或保留指定小数位。解题时要清晰展示所有步骤，因为 IGCSE 考试会按步骤给分。
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