📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程
Quadratic equations are one of the most frequently assessed topics in IGCSE Mathematics. They appear in algebra papers, in coordinate geometry questions, and in real-world modelling tasks. A solid grasp of the standard form, the three main algebraic methods, the discriminant and the graphical meaning of roots will lift your grade across several question types at once.
二次方程是 IGCSE 数学中考查频率最高的主题之一。它们出现在代数试卷、坐标几何题和实际建模问题中。扎实掌握标准形式、三种主要代数解法、判别式以及根的图像意义，能同时提升你在多种题型上的得分。
1. What is a Quadratic Equation? | 什么是二次方程？
+ b x + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic, and the whole quadratic toolkit no longer applies.
二次方程是一个次数为 2 的多项式方程。它含有最高次数为 2 的变量，并且总可以写成 a x² + b x + c = 0 的形式，其中 a、b、c 是常数且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程，整套二次方程的方法也就不再适用。
a x² + b x + c = 0, a ≠ 0
amination you may be asked to solve a quadratic equation, sketch its graph, or use it to model a practical situation such as the area of a rectangle or the path of a thrown object.
常见的例子包括 x² + 5x + 6 = 0、3x² - 2x - 1 = 0 和 2x² = 8。在 IGCSE 考试中，你可能需要解二次方程、画出它的图像，或者用它来模拟实际情境，例如长方形的面积或抛出物体的运动轨迹。
2. Standard Form and Key Vocabulary | 标准形式与关键术语
h the terms written in descending powers of x. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term. The values of x that make the equation true are called roots, or solutions, or x-intercepts when you read them off a graph.
标准形式为 a x² + b x + c = 0，各项按 x 的降幂排列。ax² 项称为二次项，bx 称为一次项，c 称为常数项。使方程成立的 x 值称为根，也叫解；如果从图像上读取，则称为 x 轴截距。
- The coefficient a controls the width and the direction of the parabola: a > 0 opens upwards, a < 0 opens downwards.
系数 a 决定抛物线的开口宽窄与方向：a > 0 开口向上，a < 0 开口向下。 - The constant c is the y-intercept, because substituting x = 0 gives y = c.
常数项 c 就是 y 轴截距，因为代入 x = 0 得到 y = c。 - A quadratic equation has at most two real roots, but it may have one repeated root or none at all.
二次方程最多有两个实根，但也可能有一个重根或没有实根。
3. Rearranging into Standard Form | 整理成标准形式
t be written as x² - 5x + 6 = 0. Solve only after this step, because the values of a, b and c read straight off an unrearranged equation will usually be wrong.
在使用任何代数方法之前，先把方程整理为所有项在一边、另一边为零的形式。例如 x² = 5x - 6 应先写成 x² - 5x + 6 = 0。一定要在这一步之后才求解，因为直接从未整理的方程上读取 a、b、c 的数值通常是错的。
x = 5 and x - 4 = 5 separately, which is invalid. Likewise, (x + 1)² = 9 is best handled by expanding or by taking square roots on both sides, not by removing the square prematurely.
要留意隐藏的二次方程和括号。方程 x(x - 4) = 5 必须先展开成 x² - 4x - 5 = 0 再因式分解；常见错误是分别令 x = 5 和 x - 4 = 5，这是无效的。同样，(x + 1)² = 9 最好通过展开或两边同时开平方来处理，不能过早去掉平方。
4. Method 1: Factorisation | 方法一：因式分解
and whose sum is b. For x² + 5x + 6 = 0, the numbers 2 and 3 have product 6 and sum 5, so the equation becomes (x + 2)(x + 3) = 0, giving x = -2 or x = -3.
当根是整数或简单分数时，因式分解是最快的方法。对于 a = 1 的 x² + b x + c = 0，要找两个数，使它们的乘积等于 c、和等于 b。对于 x² + 5x + 6 = 0，2 和 3 的乘积是 6、和是 5，所以方程变为 (x + 2)(x + 3) = 0，得到 x = -2 或 x = -3。
The zero product rule justifies the final step: if the product of two factors is zero, at least one factor must be zero. This is why the entire method depends on having zero on one side of the equation.
最后一步的依据是零积法则：如果两个因式的乘积为零，那么至少有一个因式为零。这正是为什么整个方法都依赖于方程的一边为零。
t the middle term and factorise in pairs. 3x² - 3x + x - 1 = 3x(x - 1) + 1(x - 1) = (3x + 1)(x - 1), so x = -1/3 or x = 1.
当 a ≠ 1 时，例如 3x² - 2x - 1 = 0，可用 ac 法：先算 a 与 c 的乘积得 -3，再找两个数使其乘积为 -3、和为 -2，即 -3 和 +1，然后拆开中间项并分组分解。3x² - 3x + x - 1 = 3x(x - 1) + 1(x - 1) = (3x + 1)(x - 1)，所以 x = -1/3 或 x = 1。
Note the sign rule that saves many marks: in (x + p)(x + q), the roots are -p and -q, not p and q. Always reverse the sign of the number inside the bracket.
记住一个能救回很多分的符号规则：在 (x + p)(x + q) 中，根是 -p 和 -q，而不是 p 和 q。括号内数字的符号一定要取反。
5. Method 2: Completing the Square | 方法二：配方法
+ c, take half of b, square it, and add and subtract that value. For x² + 6x + 1 = 0 the half of 6 is 3 and 3² = 9, so x² + 6x + 1 = (x + 3)² - 9 + 1 = (x + 3)² - 8.
配方法把二次式改写成 (x + p)² = q 的形式，这也是该表达式的顶点式。对 x² + b x + c 配方时，取 b 的一半、平方，再加上并减去这个值。对 x² + 6x + 1 = 0，6 的一半是 3，而 3² = 9，所以 x² + 6x + 1 = (x + 3)² - 9 + 1 = (x + 3)² - 8。
(x + 3)² = 8 → x = -3 ± √8 = -3 ± 2√2
If a is not 1, first divide the whole equation by a, or factor a out of the first two terms before completing the square. This method is the one to choose when the question says "by completing the square", when you need the exact form of an irrational answer, or when you are asked for the coordinates of the turning point.
如果 a 不等于 1，先把整个方程除以 a，或者在配方前把 a 从前两项中提取出来。当题目明确要求「用配方法求解」、需要无理根的精确形式，或者需要求顶点坐标时，就应选择这个方法。
t shows immediately that the minimum or maximum of the expression occurs at x = -b / (2a).
配方法同时也是求根公式的证明来源，并且能立刻看出表达式的最小值或最大值出现在 x = -b / (2a) 处。
6. Method 3: The Quadratic Formula | 方法三：求根公式
The quadratic formula solves any quadratic equation and never fails, which makes it the safest fallback in the exam. Substitute the values of a, b and c from the standard form, keeping the signs, and simplify carefully.
求根公式可以解任何二次方程，几乎不会失效，因此在考试中它是最稳妥的兜底方法。把标准形式中的 a、b、c（连同符号）代入，然后仔细化简。
x = ( -b ± √b² - 4ac ) / (2a)
a = 3, b = -2 and c = -1. Then b² - 4ac = (-2)² - 4 × 3 × (-1) = 4 + 12 = 16, so x = ( 2 ± √16 = 1 or x = -1/3. Both roots match the factorisation result, which is a useful self-check.
以 3x² - 2x - 1 = 0 为例，a = 3、b = -2、c = -1。则 b² - 4ac = (-2)² - 4 × 3 × (-1) = 4 + 12 = 16，所以 x = ( 2 ± √16 ) / 6 = ( 2 ± 4 ) / 6，得到 x = 1 或 x = -1/3。两个根与因式分解的结果一致，这是一个很好的自查方法。
- Write the formula down first, then substitute, so a sign slip costs one line rather than the whole question.
先把公式写下来再代入，这样一个符号失误只损失一行，而不是整道题。 - Keep b² and the product 4ac separate before subtracting; most errors come from combining them mentally.
把 b² 与乘积 4ac 分开写出再相减；绝大多数错误都来自心算合并。 - If the answer is irrational, leave it in surd form unless the question specifies a number of decimal places or significant figures.
若答案是无理数，除题目指定小数位或有效数字外，应保留根式形式。
7. The Discriminant | 判别式
The expression b² - 4ac, usually written as Δ, is called the discriminant. It tells you how many real roots a quadratic equation has before you do any solving, and it is examined directly in IGCSE as a standalone question.
表达式 b² - 4ac 通常记作 Δ，称为判别式。它能在求解之前就告诉你二次方程有多少个实根，在 IGCSE 中常作为独立小题直接考查。
| Value of Δ | Δ 的值 | Number of real roots | 实根个数 | Graph | 图像 |
| Δ > 0 | Two distinct real roots | 两个不同实根 | Cuts the x-axis twice | 与 x 轴交于两点 |
| Δ = 0 | One repeated root | 一个重根 | Touches the x-axis once | 与 x 轴相切于一点 |
| Δ < 0 | No real roots | 无实根 | Does not meet the x-axis | 与 x 轴不相交 |
-2. By contrast x² + x + 1 has Δ = 1 - 4 = -3 < 0, so it has no real solutions at all.
例如 x² + 4x + 4 的 Δ = 16 - 16 = 0，因此它在 x = -2 处有一个重根。相反，x² + x + 1 的 Δ = 1 - 4 = -3 < 0，所以它完全没有实数解。
A frequent exam phrasing is "find the value of k for which the equation has equal roots" 4ac = 0, substitute the expressions containing k, and solve the resulting equation for k.
考试中常见的问法是「求使方程有两个相等实根的 k 值」。此时令 b² - 4ac = 0，代入含 k 的表达式，再解出 k。
8. The Graphical Meaning of Roots | 根的图像意义
coordinates where this curve crosses the x-axis, because those are the points where y = 0.
y = a x² + b x + c 的图像是一条抛物线。方程 a x² + b x + c = 0 的根正好就是这条曲线与 x 轴交点的横坐标，因为这些点处 y = 0。
Axis of symmetry: x = -b / (2a)
-coordinate, and therefore the minimum or maximum value of the expression, is found by substituting this x-value back into the equation. The y-intercept is always c.
顶点（即转折点）位于对称轴 x = -b / (2a) 上。它的纵坐标，也就是该表达式的最小值或最大值，可以通过把这个 x 值代回原式求得。y 轴截距始终是 c。
Graph sketching questions usually expect four features: the correct shape (upward or downward), the y-intercept, the x-intercept(s) if they exist, and the coordinates of the turning point. If Δ < 0 the curve simply floats above or below the axis and never crosses it.
作图题通常要求四个要素：正确的开口方向、y 轴截距、x 轴截距（若存在）以及顶点坐标。如果 Δ < 0，曲线只是悬在 x 轴上方或下方，从不与之相交。
9. Forming Equations from Word Problems | 由应用题建立方程
Modelling questions ask you to translate words into a quadratic equation, solve it, and then interpret the answer in context. The algebra is usually the easy part; the mark scheme rewards a clearly defined variable and a rejected impossible root.
建模题要求你把文字翻译成二次方程、解出它，再结合情境解释答案。代数运算通常是最容易的部分；评分标准真正看重的是明确设定变量，以及舍去不合理的根。
- 40 = 0, that is (x + 8)(x - 5) = 0. The roots are x = -8 and x = 5, but a length cannot be negative, so the width is 5 cm and the length is 8 cm.
例如一个长方形，长比宽多 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3，由 x(x + 3) = 40 得 x² + 3x - 40 = 0，即 (x + 8)(x - 5) = 0。根为 x = -8 和 x = 5，但长度不可能为负，所以宽为 5 cm、长为 8 cm。
- Always define the variable in words before writing the equation.
在列方程之前，务必用文字定义变量。 - Check whether the context makes one root meaningless: negative lengths, negative times and non-integer counts of people are all rejected.
检查情境是否使其中一个根无意义：负长度、负时间以及非整数的人数都应舍去。 - Finish with a sentence stating the answer with units, not just the value of x.
最后用一句话写出带单位的答案，而不是只写 x 的值。
10. Exam Technique and Common Mistakes | 考试技巧与常见错误
Most lost marks in quadratic questions come from routine slips rather than from not knowing the method. Reading the question twice and checking the sign of each coefficient removes a large share of them.
二次方程题上丢的分，大多来自常规失误，而不是不会方法。把题目读两遍、逐个检查系数的符号，就能避免其中很大一部分。
- getting to rearrange to = 0 before factorising, which usually destroys the whole solution.
因式分解前忘记整理成 = 0 的形式，这通常会毁掉整道题的解答。 - Dividing both sides by x, which loses the root x = 0 and therefore loses marks.
方程两边同时除以 x，会丢掉根 x = 0，从而失分。 - Dropping the minus sign when b is negative, especially inside b² - 4ac.
当 b 为负数时漏掉负号，尤其是在 b² - 4ac 中。 - Rounding too early in a multi-step calculation; keep full accuracy until the final line.
在多步计算中过早取整；应保持完整精度直到最后一步。
Show the substitution, show the factorisation or the formula, and always write both roots as a pair. A final check by substituting each root back into the original equation takes seconds and confirms the answer.
写出代入过程，写出因式分解或公式，并且始终把两个根成对写出。最后把每个根代回原方程检验只需几秒钟，却能确认答案的正确性。
11. Quick Reference Summary | 速查总结
Use this summary as a last-minute revision sheet. If you can reproduce each line from memory, you are ready for the quadratic questions in your paper.
把这份总结当作考前速查表。如果你能凭记忆复现每一行，就说明你已经准备好应对试卷中的二次方程题了。
Standard form: a x² + b x + c = 0, a ≠ 0
Formula: x = ( -b ± √b² - 4ac ) / (2a)
Discriminant: Δ = b² - 4ac
Axis of symmetry and vertex x-coordinate: x = -b / (2a)
- Three methods: factorisation, completing the square, quadratic formula.
三种解法：因式分解、配方法、求根公式。 - Δ > 0 gives two roots, Δ = 0 gives one repeated root, Δ < 0 gives no real roots.
Δ > 0 有两个根，Δ = 0 有一个重根，Δ < 0 没有实根。 - Roots, solutions and x-intercepts are three names for the same values.
根、解与 x 轴截距是同一批数值的三种叫法。 - Always rearrange to standard form first, and always reject roots that the context forbids.
务必先整理成标准形式，并且始终舍去情境不允许的根。
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