📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解二次方程
Quadratic equations are one of the most heavily examined topics in IGCSE Mathematics. They appear as straightforward "solve for x" questions, as part of graph sketching, inside geometry problems about areas, and as the algebra engine behind many word problems. Whichever board you sit - Cambridge 0580, Edexcel, Oxford AQA or OCR - you are expected to recognise a quadratic, rearrange it into standard form, and choose the fastest reliable method to solve it. This guide covers every method on the syllabus: factorising, completing the square and the quadratic formula, plus the discriminant, the link to graphs, worked word problems and the mistakes that cost marks every year.
二次方程是 IGCSE 数学中考查频率最高的知识点之一。它可能以直接的 "解方程求 x" 形式出现，也可能藏在图像作图、几何面积计算和应用题当中。无论你参加的是剑桥 0580、爱德思、牛津 AQA 还是 OCR 考试，都需要能够识别二次方程、把它整理成标准形式，并选择最快、最可靠的方法求解。本文系统讲解考纲中的每一种解法：因式分解法、配方法和求根公式，并进一步讲解判别式、与图像的联系、应用题范例，以及每年都在大量失分的常见错误。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree two, which simply means that the highest power of the unknown that appears is 2. Every quadratic equation can be rearranged into the standard form shown below, where a, b and c are constants and a is not zero. If a were zero, the x² term would disappear and the equation would collapse into a linear equation, so the condition a ≠ 0 is part of the definition.
二次方程是次数为 2 的多项式方程，也就是说未知数的最高次数是 2。任何一个二次方程都可以整理成下面所示的标准形式，其中 a、b、c 是常数，且 a 不能为零。如果 a 等于零，x² 项就会消失，方程就退化为一次方程，因此 a ≠ 0 是定义的一部分。
ax² + bx + c = 0, a ≠ 0
ecause those are the points where y = 0. This algebra-graph link is worth remembering: it lets you check an answer by sketching, and it explains why a quadratic may have two roots, one repeated root, or no real roots at all.
函数 y = ax² + bx + c 的图像是一条抛物线。解方程 ax² + bx + c = 0，实际上就是求这条抛物线与 x 轴交点的横坐标，因为这些点恰好满足 y = 0。这个"代数与图像"的联系非常值得记住：它能让你用草图检验答案，也解释了为什么二次方程可能有两个根、一个重根，或者根本没有实数根。
c: 3x² − 7 = 0 is quadratic even though there is no x term (b = 0), and x² = 5x is quadratic even though there is no constant term (c = 0) once it is rearranged. By contrast, 2x + 5 = 0 is linear, and x³ − 4x = 0 is cubic, not quadratic.
有些二次方程会"伪装"自己，所以在选择方法之前一定要仔细读题。方程并不需要每一项都是二次的：3x² − 7 = 0 中虽然没有 x 项（b = 0），但它仍是二次方程；x² = 5x 整理后没有常数项（c = 0），同样是二次方程。相反，2x + 5 = 0 是一次方程，x³ − 4x = 0 是三次方程，都不是二次方程。
- Quadratic: x² − 9 = 0, 4x² + 3x = 0, (x + 1)(x − 3) = 5 (expand first), 2(x − 1)² = 8.
- Not quadratic: 3x + 1 = 0, x³ = 8, 1/x + x = 2 (this is not a polynomial equation).
- 二次方程：x² − 9 = 0、4x² + 3x = 0、(x + 1)(x − 3) = 5（需先展开）、2(x − 1)² = 8。
- 不是二次方程：3x + 1 = 0、x³ = 8、1/x + x = 2（它不是多项式方程）。
2. Standard Form and Coefficients | 标准形式与系数
la or the discriminant method, and it is where most careless marks are lost. In 2x² + 3x − 5 = 0 we have a = 2, b = 3 and c = −5.
字母 a、b、c 称为二次方程的系数。正确识别这些系数，是使用求根公式或判别式的第一步，也是最多"粗心失分"发生的地方。在 2x² + 3x − 5 = 0 中，a = 2，b = 3，c = −5。
2x² + 3x − 5 = 0
a = 3, b = 1 and c = −4. Note that b = 1, not 0: an x term with no visible number still counts as 1x. Notice also that the coefficient of x becomes positive after the rearrangement, which immediately removes a common source of sign errors.
在读取系数之前，一定要先把方程整理成标准形式。例如 4 − x = 3x² 应整理为 3x² + x − 4 = 0，因此 a = 3，b = 1，c = −4。注意 b = 1 而不是 0：省略数字的 x 项其实系数是 1。同时注意整理之后 x 的系数变成正数，这立刻消除了一类常见的符号错误。
3x² + x − 4 = 0
| Equation as given | 原方程 | Standard form | 标准形式 | a | b | c |
| 2x²
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