📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：求解一元二次方程
Quadratic equations are one of the most heavily examined topics in IGCSE Mathematics (0580 / 0980). They appear in algebra questions, in graph work, in problem-solving papers and in questions that combine several skills at once. This revision guide brings the essential methods together in one place: recognising the standard form, rearranging equations, factorising, completing the square, using the quadratic formula, interpreting the discriminant, and reading roots from a quadratic graph.
一元二次方程是 IGCSE 数学（0580 / 0980）考查最密集的专题之一，它既出现在代数运算题中，也出现在函数图像题、应用题，以及需要综合多项技能的综合题里。本复习指南把这些核心方法集中在一起：识别标准形式、整理方程、因式分解、配方法、使用求根公式、理解判别式，以及从二次函数图像上读出方程的根。
Work through the sections in order, then use the final checklist for a fast revision pass the night before the exam.
建议按顺序学习每一节，考前最后一晚再用文末清单做一次快速回顾。
1. What Is a Quadratic Equation? | 什么是一元二次方程？
stants and a ≠ 0. The highest power of x is 2, and that single fact gives the equation its name. If the highest power were 1, the equation would be linear, not quadratic.
一元二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。未知数 x 的最高次数是 2，正是这一点决定了"二次"这个名称。如果最高次数是 1，那它就是一次方程，而不是二次方程。
ax² + bx + c = 0, a ≠ 0
re solved by simple rearrangement instead. Some equations may need rearranging before they show the standard form, which is covered in section 3.
常见例子包括 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 x² − 9 = 0。而像 3x − 2 = 0 或 4x + 7 = 0 这样的方程是一次方程，只需简单移项即可求解。有些方程需要先整理才能变成标准形式，这一点在第 3 节详细说明。
A quadratic equation normally has two solutions, called roots. In some special cases the two roots are equal, and in others there are no real roots at all — the discriminant in section 7 tells you which situation you are in.
一元二次方程通常有两个解，称为根。在特殊情况下两个根相等，有时则完全没有实数根——第 7 节的判别式会告诉你属于哪一种情况。
2. Standard Form and Key Terms | 标准形式与关键术语
he linear term, and c is the constant term. The coefficient a must not be zero, but b or c — or both — = 0 (where b = 0) and 3x² + 5x = 0 (where c = 0) are perfectly valid quadratic equations.
标准形式为 ax² + bx + c = 0。其中 ax² 称为二次项，bx 称为一次项，c 称为常数项。二次项系数 a 不能为 0，但 b 或 c（甚至两者同时）都可以为 0。因此 x² − 9 = 0（b = 0）和 3x² + 5x = 0（c = 0）都是完全合法的二次方程。
| Term | 项 | Name | 名称 | Example in 2x² + 3x − 2 = 0 |
| ax² | Quadratic term | 二次项 | 2x² (a = 2) |
| bx | Linear term | 一次项 | 3x (b = 3) |
| c | Constant term | 常数项 | −2 (c = −2) |
Before you choose a method, always write down the values of a, b and c with their signs. A very large number of lost marks in this topic come from dropping a minus sign at this first step.
在选择题型之前，先把 a、b、c 的数值连同符号写清楚。这个专题中大量失分，都是因为第一步就丢掉了负号。
3. Rearranging into Standard Form | 整理成标准形式
Never apply the quadratic formula to an equation that is not yet in standard form. Expand brackets, clear fractions, multiply by the denominator, then move every term to one side so that the other side is exactly zero.
千万不要对尚未化成标准形式的方程直接套用求根公式。要先展开括号、消去分母、两边同乘，然后把所有项移到同一侧，使另一侧恰好为 0。
- 3x² = 5x + 2 → 3x² − 5x − 2 = 0 (a = 3, b = −5, c = −2) | 两边同时减去 5x 和 2 即可。
- (x + 1)(x − 3) = 5 → x² − 2x − 3 = 5 → x² − 2x − 8 = 0 | 先展开，再把 5 移到左边。
- (x + 2)² = 9 → x² + 4x + 4 = 9 → x² + 4x − 5 = 0 | 展开完全平方后再移项。
- x + 6 ÷ x = 5, x ≠ 0 → x² + 6 = 5x → x² − 5x + 6 = 0 | 两边同乘 x，注意注明 x ≠ 0。
rm by the denominator first. Keeping the equation tidy at this stage makes factorising and substitution far more reliable later.
如果方程中出现类似 (2x + 1) ÷ 3 = x² 的分数，先两边同乘分母。此时把式子整理干净，后面的因式分解和代入计算就会可靠得多。
4. Solving by Factorising | 因式分解法
Factorising is the fastest method and is usually the one examiners expect when the numbers are small and friendly. It relies on the zero product property: if two expressions multiply to give zero, at least one of them must itself be zero.
因式分解是最快的方法，当数字较小、比较友好时，考官通常就是希望你用这一招。它依据的是零乘积性质：若两个式子的乘积为 0，则其中至少有一个式子本身等于 0。
If (x − p)(x − q) = 0, then x = p or x = q
because (−2) × (−3) = 6 and (−2) + (−3) = −5. So (x − 2)(x − 3) = 0 gives x = 2 or x = 3.
对 a = 1 的 x² + bx + c = 0，要找两个数，使它们的乘积等于 c、和等于 b。对 x² − 5x + 6 = 0，这一对数是 −2 和 −3，因为 (−2) × (−3) = 6，且 (−2) + (−3) = −5。于是 (x − 2)(x − 3) = 0，得 x = 2 或 x = 3。
so the numbers are 4 and −1. Splitting gives 2x² + 4x − x − 2 = 2x(x + 2) − 1(x + 2) = (2x − 1)(x + 2), so x = ½ or x = −2.
当 a ≠ 1 时，可以使用 ac 法：找两个数，使其乘积为 ac、和为 b，把中间项拆开，再分组提取公因式。对 2x² + 3x − 2 = 0，ac = −4，b = 3，所以这两个数是 4 和 −1。拆项得 2x² + 4x − x − 2 = 2x(x + 2) − 1(x + 2) = (2x − 1)(x + 2)，因此 x = ½ 或 x = −2。
and perfect squares such as x² + 6x + 9 = (x + 3)², which gives the repeated root x = −3.
两个特殊结构能省下大量时间：平方差 x² − 9 = (x + 3)(x − 3)，以及完全平方，如 x² + 6x + 9 = (x + 3)²，它给出重根 x = −3。
5. Completing the Square | 配方法
he method to use when the question asks for an exact answer in surd form, when you need the vertex of a graph, or when factorising fails because the roots are irrational.
配方法把二次式改写为 (x + p)² = q 的形式。当题目要求以根号形式给出精确答案、需要求抛物线顶点，或者因为根是无理数而无法因式分解时，就应该用这个方法。
(x + p)² = q then x = −p ± √q
+ 5 = 0, that is (x + 3)² = 4. Taking square roots gives x + 3 = ±2, so x = −1 or x = −5.
对 x² + bx + c 配方时，先把一次项系数取一半再平方：x² + bx = (x + b ÷ 2)² − (b ÷ 2)²。例如 x² + 6x + 5 = 0 可写成 (x + 3)² − 9 + 5 = 0，即 (x + 3)² = 4。两边开平方得 x + 3 = ±2，所以 x = −1 或 x = −5。
3 = 0, then 2[(x − 2)² − 4] + 3 = 0, which simplifies to 2(x − 2)² = 5 and finally x = 2 ± √5 ÷ 2. Remember to take the square root of both sides, and always keep the ± symbol.
当 a ≠ 1 时，先把系数 a 提取出来。对 2x² − 8x + 3 = 0，写成 2(x² − 4x) + 3 = 0，再写成 2[(x − 2)² − 4] + 3 = 0，化简得 2(x − 2)² = 5，最终 x = 2 ± √5 ÷ 2。记得两边都要开平方，并且不要漏掉 ± 号。
6. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation, including those that cannot be factorised. Once the equation is in standard form, substitute a, b and c directly into the formula.
求根公式可以解任何一元二次方程，包括无法因式分解的那些。只要方程已化为标准形式，就把 a、b、c 直接代入公式。
x = (−b ± √b² − 4ac) ÷ 2a
ere a = 2, b = 3 and c = −2, so b² − 4ac = 3² − 4 × 2 × (−2) = 9 + 16 = 25. Then x = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4, giving x = ½ or x = −2 — exactly the same roots as factorising produced.
例题：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2，所以 b² − 4ac = 3² − 4 × 2 × (−2) = 9 + 16 = 25。于是 x = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4，得 x = ½ 或 x = −2，与因式分解得到的根完全一致。
pute b² − 4ac as a separate line before you reach for the square root, because most errors happen inside that expression. Give non-exact answers to three significant figures unless the paper says otherwise, and simplify surds where possible.
两个习惯能保护你的准确率。第一，当 b 为负数时把它加括号：若 b = −5，则 −b = +5。第二，把 b² − 4ac 单独作为一行先算出来，因为绝大多数错误都发生在这个式子里。若答案不是精确值，除题目另有说明外，保留三位有效数字，并尽可能把根式化简。
7. The Discriminant | 判别式
The discriminant is the expression b² − 4ac, often written as Δ. It sits underneath the square root in the quadratic formula, so its sign decides how many real roots the equation has — without you having to solve it at all.
判别式就是 b² − 4ac，通常记作 Δ。它位于求根公式的根号之下，因此它的符号直接决定方程有多少个实数根——你根本不需要真的去解方程。
| Value | 取值 | Roots | 根的个数 | Graph | 图像 |
| Δ > 0 | Two distinct real roots | 两个不相等实根 | Crosses the x-axis twice | 与 x 轴交于两点 |
| Δ = 0 | One repeated root | 一个重根 | Touches the x-axis | 与 x 轴相切 |
| Δ < 0 | No real roots | 无实数根 | Does not meet the x-axis | 与 x 轴不相交 |
5 = 0 has Δ = 4 − 20 = −20 < 0, so it has no real roots. A typical exam question asks for the values of k for which x² + kx + 9 = 0 has equal roots: set Δ = k² − 36 = 0, so k = 6 or k = −6.
如果 Δ 是完全平方数（且 a、b、c 为整数），那么根是有理数，说明这个二次式本来是可以因式分解的。例如 x² + 2x + 5 = 0 的 Δ = 4 − 20 = −20 < 0，因此没有实数根。典型考题会问：当 k 取何值时 x² + kx + 9 = 0 有相等的根？令 Δ = k² − 36 = 0，得 k = 6 或 k = −6。
8. Quadratic Graphs and Roots | 二次函数图像与根
e equation ax² + bx + c = 0 are exactly the x-coordinates where the curve cuts the x-axis.
y = ax² + bx + c 的图像是一条抛物线。若 a > 0，开口向上，形如 U；若 a < 0，开口向下，形如倒 U。方程 ax² + bx + c = 0 的根，正好就是曲线与 x 轴交点的横坐标。
he roots are x = 1 and x = 5, the line of symmetry is x = 3 and the vertex has x-coordinate 3. Substituting that value back into the equation gives the y-coordinate of the vertex.
顶点（转折点）落在对称轴 x = −b ÷ 2a 上。由于抛物线关于对称轴对称，顶点正好位于两个根的正中间：若两根为 x = 1 和 x = 5，则对称轴为 x = 3，顶点的横坐标就是 3。把这个值代回原式，即可得到顶点的纵坐标。
y = c. Sketch questions usually want the shape, the y-intercept, the roots and the vertex — four pieces of information that together earn full marks.
常数项 c 就是 y 轴截距，因为令 x = 0 得 y = c。作图题通常要求给出开口方向、y 轴截距、根以及顶点这四项信息，写全即可拿满分。
9. Choosing the Best Method | 如何选择最合适的方法
Marks are lost when students
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