📚 Mastering Quadratic Equations | 掌握二次方程
Quadratic equations run through the whole IGCSE and A-Level Mathematics syllabus: factorising simple trinomials, completing the square, applying the quadratic formula and reading key features from a parabola. This article builds the topic step by step, from the definition of a quadratic to discriminants, graphs and exam technique, so that you can recognise the question type quickly and solve it with confidence.
二次方程贯穿 IGCSE 与 A-Level 数学的整个课程：简单的三项式因式分解、配方法、二次求根公式，以及从抛物线图像中读取关键信息。本文从二次方程的定义出发，循序渐进地覆盖判别式、图像与应试技巧，帮助你快速识别题型并自信作答。
1. What Is a Quadratic Equation? | 什么是二次方程？
where a, b and c are constants and a ≠ 0. If a were zero the equation would become linear rather than quadratic, so the condition a ≠ 0 is part of the definition, not a decoration.
二次方程是未知量最高次幂为 2 的多项式方程，它总能写成一般形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。如果 a 为 0，方程就退化为一次方程而不是二次方程，因此 a ≠ 0 是定义的一部分，而不是可有可无的附加条件。
The word "quadratic" comes from the Latin for "square", because the leading term is a square term. Notice that a quadratic equation is an equation, so it always contains an equals sign and asks for the value(s) of x that make both sides balance. An expression such as x² - 5x + 6 is not an equation; it can be factorised and graphed, but it does not have "roots" until you set it equal to something.
"Quadratic" 一词源自拉丁语中的"平方"，因为最高次项是平方项。要注意，二次方程是一个方程，所以它必然含有等号，并要求求出使两边相等的 x 值。像 x² - 5x + 6 这样的表达式不是方程：它可以被因式分解、可以画图像，但在你把它设为等于某个值之前，它并没有"根"。
2. Standard Form and Coefficients | 标准形式与系数
he names of the coefficients: a is the coefficient of x², b is the coefficient of x, and c is the constant term. Identifying a, b and c correctly is essential before using the quadratic formula or the discriminant, because almost every careless error in this topic starts with a misread coefficient.
标准形式 ax² + bx + c = 0 明确了系数的名称：a 是 x² 的系数，b 是 x 的系数，c 是常数项。在使用二次公式或判别式之前，正确识别 a、b、c 至关重要，因为这一专题中几乎所有粗心错误都始于系数读错。
x - x² = 6, rearrange to -x² + 4x - 6 = 0, so a = -1, b = 4 and c = -6. Never drop a negative value of a: it controls both the shape of the parabola and the sign inside the formula.
例如在 3x² - 7x + 2 = 0 中，a = 3，b = -7，c = 2。符号要紧跟系数，因为这一项是 -7x，所以 b 是负数。类似地，在 4x - x² = 6 中要整理为 -x² + 4x - 6 = 0，因此 a = -1，b = 4，c = -6。不要忽略 a 的负号：它既决定抛物线的开口方向，也影响公式中的符号。
- Write the equation in the form ax² + bx + c = 0 first. 先写成 ax² + bx + c = 0 的形式。
- Keep negative signs attached to the correct coefficient. 负号要紧跟对应的系数。
- Never ignore a negative value of a. 不要忽略 a 的负值。
- Check whether the question wants exact surd form, decimals or both. 看清题目要求精确根式、小数还是两者都要。
3. Rearranging Into Standard Form | 整理成标准形式
ify the standard form and choose a suitable solving method. Common disguises include x² = 5x - 6, which becomes x² - 5x + 6 = 0, and 2x(x - 3) = 8, which expands to 2x² - 6x - 8 = 0 and then simplifies to x² - 3x - 4 = 0.
考题常常把二次方程"藏"起来。你的第一步是把所有项移到一边，使另一边等于 0。只有整理成标准形式后，你才能识别系数并选择适当的求解方法。常见的伪装形式包括 x² = 5x - 6，整理后为 x² - 5x + 6 = 0；以及 2x(x - 3) = 8，展开为 2x² - 6x - 8 = 0，再化简为 x² - 3x - 4 = 0。
Two habits protect your marks here. First, expand every bracket and collect like terms before you do anything else - never factorise a product that still contains a bracket in disguise. Second, if every coefficient shares a common factor, divide through by it; x² - 3x - 4 is far easier to handle than 2x² - 6x - 8, and dividing does not change the roots.
有两个习惯可以保住分数。第一，先展开所有括号并合并同类项，再做其他运算——不要把仍含括号的乘积直接因式分解。第二，如果所有系数有公因数，就先整体约去：x² - 3x - 4 比 2x² - 6x - 8 好处理得多，而且除以公因数不会改变根。
4. Solving by Factorising | 因式分解法求解
uct is ac and whose sum is b, or use the simpler case where a = 1, where the numbers multiply to c and add to b. Then use the zero-product rule: if two factors multiply to give zero, at least one of them must be zero.
当二次方程有整数根时，因式分解通常是最快的方法。整理成标准形式后，寻找两个数，使它们相乘等于 ac 且相加等于 b；当 a = 1 时更简单，只需找到相乘等于 c 且相加等于 b 的两个数。然后使用零乘积法则：如果两个因式相乘为零，那么至少有一个因式为零。
or to zero gives x = 2 or x = 3. Notice how the signs do the work: both numbers are negative because the sum is negative while the product is positive.
求解 x² - 5x + 6 = 0。我们需要两个数，乘积为 +6，和为 -5：这两个数是 -2 和 -3。因此 (x - 2)(x - 3) = 0，令每个因式为零，得到 x = 2 或 x = 3。注意符号的作用：因为和是负数而积是正数，所以两个数都是负数。
3 = 0, then factorise in pairs: 2x(x + 1) + 3(x + 1) = 0, giving (x + 1)(2x + 3) = 0, so x = -1 or x = -3/2. Always expand your answer mentally to check it returns to the original expression.
当 a 不为 1 时，例如 2x² + 5x + 3 = 0，需要两个数乘积为 2 × 3 = 6、和为 5，即 2 和 3。拆开中间项：2x² + 2x + 3x + 3 = 0，然后分组分解：2x(x + 1) + 3(x + 1) = 0，得到 (x + 1)(2x + 3) = 0，所以 x = -1 或 x = -3/2。作答后一定要在脑中展开验算，确认能还原原式。
- Product = ac, sum = b is the universal rule for splitting the middle term. 乘积 = ac、和 = b 是拆中间项的通用法则。
- Never divide both sides by x; you would lose the root x = 0. 不要两边同除以 x，否则会丢掉 x = 0 这个根。
- Always set each factor equal to zero separately. 每个因式都要分别令其为零。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² + q, which reveals the vertex of the parabola and allows you to solve any quadratic, even when the roots are irrational. The method works because x² + bx can be turned into a perfect square by adding and subtracting (b/2)².
配方法把二次式改写成 (x + p)² + q 的形式，这既能直接读出抛物线的顶点，也能求解任何二次方程，即使根是无理数。其原理在于：只要加上并减去 (b/2)²，x² + bx 就能配成完全平方。
x² + bx + c = (x + b/2)² - (b/2)² + c
, so write (x + 3)² - 9 + 1 = 0, that is (x + 3)² = 8. Taking square roots gives x + 3 = ±√8, so x = -3 ± 2√2, which is about -0.17 or -5.83. When a is not 1, factor a out of the first two terms first, then complete the square inside the bracket.
例如求解 x² + 6x + 1 = 0。6 的一半是 3，因此写成 (x + 3)² - 9 + 1 = 0，即 (x + 3)² = 8。开平方得 x + 3 = ±√8，所以 x = -3 ± 2√2，约为 -0.17 或 -5.83。当 a 不为 1 时，先把 a 从前两项提出，再在括号内配方。
p is the axis of symmetry. This is why completing the square appears so often in graph questions asking for the minimum value or the coordinates of the vertex.
在顶点形式 y = a(x + p)² + q 中，顶点坐标是 (-p, q)，直线 x = -p 是对称轴。这就是配方法在求最小值或顶点坐标的图像题中频繁出现的原因。
6. The Quadratic Formula | 二次公式
The quadratic formula solves every quadratic equation, whether or not it factorises. You must first write the equation in standard form, then substitute a, b and c carefully, keeping brackets around negative values.
二次公式可以求解任何二次方程，无论它能否因式分解。你必须先把方程写成标准形式，再仔细代入 a、b、c，并给负值加上括号。
x = (-b ± √b² - 4ac) / (2a)
Solve 2x² - 4x - 3 = 0. Here a = 2, b = -4 and c = -3, so x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4. Simplify √40 = 2√10, giving x = 1 ± (√10)/2, approximately 2.58 and -0.58. Leave the answer in surd form if the question says "exact value" or "in the form p ± q√r".
求解 2x² - 4x - 3 = 0。此处 a = 2，b = -4，c = -3，所以 x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4。化简 √40 = 2√10，得 x = 1 ± (√10)/2，约等于 2.58 和 -0.58。如果题目要求"精确值"或"p ± q√r 的形式"，答案应保留根式。
- Substitute with brackets: -(-4) not -4. 代入时加括号：写 -(-4) 而不是 -4。
- Simplify the surd and the fraction before rounding. 先化简根式与分数，再取近似值。
- Give both roots unless the context rejects one. 除非题意排除，否则两个根都要写。
7. The Discriminant | 判别式
The expression b² - 4ac, written as Δ, tells you how many real roots a quadratic has before you solve it. This is one of the most reliably examined facts in the whole topic.
表达式 b² - 4ac 记作 Δ，它能在你求解之前告诉你二次方程有多少个实根。这是整个专题中考查最稳定的事实之一。
| Δ = b² - 4ac | Nature of roots 根的性质 | Graph 图像 |
| Δ > 0 | Two distinct real roots 两个不同实根 | Crosses the x-axis twice 与 x 轴相交两次 |
| Δ = 0 | One repeated real root 一个重根 | Touches the x-axis 与 x 轴相切 |
| Δ < 0 | No real roots 无实根 | Does not meet the x-axis 与 x 轴不相交 |
s x² + kx + 9 = 0, and asks for the values of k that give equal roots: set Δ = k² - 36 = 0, so k = 6 or k = -6. Remember that equal roots means a perfect square, and that a positive perfect-square discriminant also guarantees rational roots.
例如 x² + 4x + 5 = 0 的 Δ = 16 - 20 = -4 < 0，因此没有实数解。典型考题会给出含未知常数的二次方程，如 x² + kx + 9 = 0，要求求出使方程有等根的 k 值：令 Δ = k² - 36 = 0，得 k = 6 或 k = -6。要记住：等根意味着完全平方；判别式为正的完全平方数时，根也是有理数。
8. Graphs of Quadratic Functions | 二次函数的图像
ntercept is c, found by setting x = 0, and the x-intercepts are the roots of ax² + bx + c = 0.
y = ax² + bx + c 的图像是抛物线。若 a > 0，开口向上且有最小值点；若 a < 0，开口向下且有最大值点。令 x = 0 得 y 轴截距 c；图像与 x 轴的交点横坐标就是方程 ax² + bx + c = 0 的根。
it. Substituting that x-value back into the equation gives the y-coordinate of the vertex. Completing the square is an equally valid route, and many mark schemes accept either method as long as the coordinates are stated clearly.
对称轴是竖直线 x = -b/(2a)，顶点就在这条直线上。把该 x 值代回方程即可得到顶点的 y 坐标。配方法同样可行，只要坐标写清楚，多数评分标准都接受两种方法。
Sketch graphs are marked for shape, intercepts and turning point, not for artistic quality. Label the coordinates of any intercept you have calculated, and draw a smooth curve rather than a series of straight segments. If a question says "hence" or "use your graph", it usually wants you to read off a value or to justify the number of solutions.
草图的给分点是形状、截距与顶点，而不是画得多好看。把算出的交点坐标标出来，并画一条平滑曲线而不是一段段折线。如果题目出现"hence"或"use your graph"，通常是要你读图取值或说明解的个数。
9. Sum and Product of Roots | 根与系数的关系
dentities let you answer questions about roots without ever solving the equation, and they are a powerful way to check answers you have already found.
若 ax² + bx + c = 0 的两个根为 α 和 β，则 α + β = -b/a，αβ = c/a。这两个恒等式让你无需解方程就能回答关于根的问题，也是检验已求答案的有力工具。
hich equals (α + β)² - 2αβ, or for a new quadratic with roots α + 1 and β + 1, built from x² - (sum)x + (product) = 0.
以 2x² - 8x + 6 = 0 为例，其根为 1 和 3。则 α + β = 4 = -(-8)/2，αβ = 3 = 6/2，与恒等式完全一致。有些题目要求 α² + β²，它等于 (α + β)² - 2αβ；或者要求构造以 α + 1 和 β + 1 为根的新二次方程，可用 x² - (和)x + (积) = 0 来构造。
10. Word Problems and Modelling | 应用题与建模
Many quadratic questions are wrapped in a real-world context: the area of a rectangle, the height of a thrown ball, profit as a function of price, or two numbers related by a condition. The mathematics is the same, but the marks come from setting the equation up correctly.
许多二次方程题目被包装在真实情境中：矩形面积、抛出物体的高度、利润与价格的关系，或两个数之间的条件关系。数学本身不变，但分数来自能否正确建立方程。
ret the answer in context. For h = 20t - 5t², the ball is on the ground when h = 0, so 5t(4 - t) = 0 and t = 0 or t = 4 seconds; t = 0 is the launch and t = 4 is the landing.
按固定流程作答：先用文字定义变量，把每个条件翻译成方程，整理成标准形式，求解，最后回到情境中解释答案。对于 h = 20t - 5t²，小球落地时 h = 0，即 5t(4 - t) = 0，得 t = 0 或 t = 4 秒；t = 0 是抛出时刻，t = 4 是落地时刻。
Always test whether a root makes sense in the real world. A negative length, a negative time or a negative number of items must be rejected, and stating that rejection explicitly usually earns a mark. Give units with the final answer, and round only at the very end to avoid accumulating error.
务必检验每个根在现实中是否合理。负的长度、负的时间或负的物品数量都必须舍去，并且明确写出舍去的理由，这通常能得分。最终答案要带单位，并只在最后一步取近似值，以免误差累积。
11. Common Exam Mistakes | 常见考试错误
Most lost marks in this topic are not conceptual but mechanical. Knowing the standard traps in advance is the cheapest way to raise your grade.
这一专题中丢掉的分数大多不是概念性的，而是操作性的。提前熟悉这些常见陷阱，是提分成本最低的方法。
- Solving before rearranging into ax² + bx + c = 0. 未整理成 ax² + bx + c = 0 就急着求解。
- Using the wrong sign when a is negative, especially writing b instead of -b. 当 a 为负时符号出错，尤其把 -b 写成 b。
- Forgetting that √b² - 4ac applies to the whole expression b² - 4ac. 忘记根号覆盖的是整个 b² - 4ac。
- Giving only one root when two are required. 需要两个根时只写了一个。
- Rounding too early, so the second root drifts outside the accepted range. 过早取近似值，导致第二个根超出容许范围。
- Rejecting a valid root or keeping an impossible one in context questions. 情境题中舍掉了有效根或保留了不合理的根。
- Dividing both sides by x and losing the root x = 0. 两边同除以 x，丢掉根 x = 0。
Read the command word carefully as well: "solve", "factorise", "complete the square" and "show that" each demand a different form of answer, and a correct value written in the wrong form can still lose the final mark.
同时要看清指令词："solve"（求解）、"factorise"（因式分解）、"complete the square"（配方）与 "show that"（证明）各自要求不同的答案形式，即使数值正确，形式不对仍可能丢掉最后一分。
12. Choosing the Fastest Method | 选择最快的方法
Method choice saves time in the exam. Try factorising first when the coefficients are small and the roots look like integers; use the formula when the numbers are awkward or the answer must be exact; complete the square when the question asks for the vertex or the minimum value; read the discriminant when the question asks how many roots exist or involves an unknown constant.
方法选择能帮你在考场上省时间。系数较小、根看起来是整数时先试因式分解；数字不整齐或要求精确值时用公式；题目问顶点或最小值时用配方；题目问根的个数或含未知常数时先看判别式。
| Question cue 题目提示 | Best method 最佳方法 |
| Small integer coefficients 系数为小整数 | Factorising 因式分解 |
| Exact value or surd form 精确值或根式形式 | Quadratic formula 二次公式 |
