Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. Almost every exam paper contains at least one question on solving a quadratic equation, whether it is by factorisation, completing the square, or using the quadratic formula. This revision guide gives you a step-by-step breakdown of every method you need, along with common pitfalls and exam tips.

二次方程是 IGCSE 数学中最重要的主题之一。几乎每份考试试卷都至少包含一道关于解二次方程的题目，无论是因式分解法、配方法，还是使用二次公式。本复习指南将逐步为你拆解所需的每一种方法，并指出常见陷阱和考试技巧。

1. What Is a Quadratic Equation? | 什么是二次方程

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The name ‘quadratic’ comes from the Latin word ‘quadratus’, meaning square, because the highest power of x is 2.

二次方程是可以写成标准形式 ax² + bx + c = 0 的任何方程，其中 a、b、c 是常数，且 a ≠ 0。名称 “quadratic” 源自拉丁语 “quadratus”，意为平方，因为 x 的最高次数是 2。

Examples of quadratic equations include 2x² + 5x − 3 = 0 and x² − 9 = 0. Notice that the second example has b = 0, which makes it a special case called a ‘pure’ quadratic equation. The first example is a ‘complete’ quadratic because all three coefficients are non-zero.

二次方程的例子包括 2x² + 5x − 3 = 0 和 x² − 9 = 0。注意第二个例子的 b = 0，这是一种特殊情况，称为 “纯” 二次方程。第一个例子是 “完全” 二次方程，因为三个系数都非零。

You must always rearrange a given equation into the standard form ax² + bx + c = 0 before solving. For example, 3x² = 7x − 2 must be rewritten as 3x² − 7x + 2 = 0 first.

在求解之前，必须始终将给定方程整理成标准形式 ax² + bx + c = 0。例如，3x² = 7x − 2 必须先改写为 3x² − 7x + 2 = 0。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic has simple integer factors. The key idea is the null factor law: if the product of two expressions is zero, then at least one of the expressions must be zero. In symbols, if p × q = 0, then p = 0 or q = 0.

因式分解是当二次方程具有简单整数因子时最快的方法。关键思想是 零因子定律：如果两个表达式的乘积为零，那么至少其中一个表达式必须为零。用符号表示，如果 p × q = 0，则 p = 0 或 q = 0。

For a quadratic of the form x² + bx + c, you look for two numbers whose sum is b and whose product is c. For example, to solve x² + 7x + 12 = 0, find two numbers with sum 7 and product 12: 3 and 4. Hence (x + 3)(x + 4) = 0, giving x = −3 or x = −4.

对于形式为 x² + bx + c 的二次方程，你需要找到两个数，其和为 b，积为 c。例如，解 x² + 7x + 12 = 0，找到两个和为 7、积为 12 的数：3 和 4。因此 (x + 3)(x + 4) = 0，得到 x = −3 或 x = −4。

When the coefficient a is not 1, for example 2x² + 7x + 3 = 0, you must find factors of a × c = 6 that add to b = 7. Here 1 and 6 work, so split the middle term: 2x² + x + 6x + 3 = 0, then factor by grouping: x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0. Therefore x = −½ or x = −3.

当系数 a 不是 1 时，例如 2x² + 7x + 3 = 0，你需要找到 a × c = 6 的因子，使其相加等于 b = 7。这里 1 和 6 满足条件，因此拆中项：2x² + x + 6x + 3 = 0，然后分组因式分解：x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0。因此 x = −½ 或 x = −3。

Always check your factors by expanding them back. If the expansion does not return the original quadratic, your factorisation is wrong.

始终通过展开来检查你的因式。如果展开后不能返回原二次方程，说明你的因式分解有误。

3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This is extremely useful not only for solving equations, but also for finding the turning point of a parabola and for deriving the quadratic formula.

配方法将二次方程改写为 (x + p)² + q 的形式。这不仅对解方程非常有用，也用于求抛物线的顶点以及推导二次公式。

To complete the square for x² + bx + c, take half of b, write (x + b/2)², then subtract (b/2)² and add c. For example, x² + 6x + 2 = 0 becomes (x + 3)² − 9 + 2 = 0, so (x + 3)² = 7.

要对 x² + bx + c 配方，取 b 的一半，写成 (x + b/2)²，然后减去 (b/2)² 并加上 c。例如，x² + 6x + 2 = 0 变成 (x + 3)² − 9 + 2 = 0，因此 (x + 3)² = 7。

Now take the square root of both sides, remembering the ± sign:

现在对两边取平方根，记得加上 ± 号：

x + 3 = ±√7, so x = −3 ± √7

When a ≠ 1, first divide the whole equation by a before completing the square. For instance, 2x² − 8x + 5 = 0 becomes x² − 4x + 2.5 = 0, then (x − 2)² − 4 + 2.5 = 0, giving (x − 2)² = 1.5 and x = 2 ± √1.5.

当 a ≠ 1 时，先在整个方程两边除以 a，然后再配方。例如，2x² − 8x + 5 = 0 变为 x² − 4x + 2.5 = 0，然后 (x − 2)² − 4 + 2.5 = 0，得到 (x − 2)² = 1.5 和 x = 2 ± √1.5。

The vertex of the parabola y = (x + p)² + q is exactly the point (−p, q), which is why completing the square is the fastest way to locate the turning point.

抛物线 y = (x + p)² + q 的顶点恰好是点 (−p, q)，这就是为什么配方法是定位顶点的最快方法。

4. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation in one step, and you should memorise it. For ax² + bx + c = 0, the solutions are given by:

二次公式可以一步解出任何二次方程，你应该熟记它。对于 ax² + bx + c = 0，解由下式给出：

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, write down the values of a, b and c first. For example, in 2x² − 3x − 5 = 0, we have a = 2, b = −3, c = −5. Substitute these into the formula:

要使用此公式，先写出 a、b、c 的值。例如，在 2x² − 3x − 5 = 0 中，a = 2，b = −3，c = −5。将这些值代入公式：

x = (3 ± √((−3)² − 4 × 2 × (−5))) / (2 × 2) = (3 ± √49) / 4 = (3 ± 7) / 4

Therefore x = (3 + 7)/4 = 10/4 = 2.5 or x = (3 − 7)/4 = −4/4 = −1.

因此 x = (3 + 7)/4 = 10/4 = 2.5 或 x = (3 − 7)/4 = −4/4 = −1。

Be extremely careful with negatives. The expression (−3)² equals 9, not −9. Also, when c is negative, −4ac becomes positive, which many students forget.

处理负数时要格外小心。表达式 (−3)² 等于 9，而不是 −9。另外，当 c 为负数时，−4ac 会变成正数，很多学生会忘记这一点。

5. The Discriminant | 判别式

The expression Δ = b² − 4ac underneath the square root sign is called the discriminant. It tells you the nature of the roots without fully solving the equation.

根号下的表达式 Δ = b² − 4ac 称为 判别式。它可以在不完全解方程的情况下告诉你根的性质。

There are three cases to know for IGCSE:

在 IGCSE 中你需要知道三种情况：

Discriminant 判别式	Nature of roots 根的性质	Graph interpretation 图像含义
b² − 4ac > 0	Two distinct real roots 两个不同的实数根	Parabola crosses the x-axis twice 抛物线与 x 轴相交两次
b² − 4ac = 0	Two equal real roots (one repeated root) 两个相等的实数根（一个重根）	Parabola touches the x-axis once 抛物线与 x 轴相切一次
b² − 4ac < 0	No real roots 无实数根	Parabola does not meet the x-axis 抛物线与 x 轴不相交

For example, the equation x² + 2x + 5 = 0 has discriminant Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16, which is negative, so there are no real roots.

例如，方程 x² + 2x + 5 = 0 的判别式 Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16，为负数，因此没有实数根。

Exam questions often ask you to ‘find the range of values of k for which the equation has real roots’. In such cases, set b² − 4ac ≥ 0 and solve the resulting inequality.

考试题经常要求你 “求出使方程有实数根的 k 的取值范围”。在这种情况下，令 b² − 4ac ≥ 0，并解所得的不等式。

6. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is always a parabola. When a > 0, the parabola opens upwards and has a minimum turning point; when a < 0, it opens downwards and has a maximum turning point.

y = ax² + bx + c 的图像始终是抛物线。当 a > 0 时，抛物线开口向上，有最小值顶点；当 a < 0 时，开口向下，有最大值顶点。

The x-coordinate of the vertex is given by x = −b / 2a. You can find the y-coordinate by substituting this value of x back into the equation. Alternatively, if you have already completed the square, the vertex is obtained directly as described in Section 3.

顶点的 x 坐标由 x = −b / 2a 给出。你可以将该 x 值代回方程求出 y 坐标。或者，如果你已经完成了配方，可直接按第 3 节所述得到顶点。

The roots of the equation ax² + bx + c = 0 correspond exactly to the x-intercepts of the graph. The y-intercept is simply the value of c, since the graph crosses the y-axis when x = 0.

方程 ax² + bx + c = 0 的根恰好对应图像的 x 截距。y 截距就是 c 的值，因为图像在 x = 0 时与 y 轴相交。

When sketching a quadratic graph, mark the intercepts, the vertex, and the axis of symmetry x = −b / 2a. This fully determines the shape of the parabola.

在画二次函数草图时，标出截距、顶点和对称轴 x = −b / 2a。这就能完全确定抛物线的形状。

7. Sum and Product of Roots | 根的和与积

If a quadratic equation ax² + bx + c = 0 has roots α and β, then the sum of the roots is α + β = −b / a and the product of the roots is αβ = c / a. These relationships let you construct a quadratic from its roots without expansion.

如果二次方程 ax² + bx + c = 0 有根 α 和 β，则根的和为 α + β = −b / a，根的积为 αβ = c / a。这些关系让你无需展开即可从根构造二次方程。

For example, if the roots are 2 and −5, then the sum is −3 and the product is −10. The equation is x² − (sum)x + (product) = 0, so x² + 3x − 10 = 0.

例如，如果根是 2 和 −5，则和为 −3，积为 −10。方程为 x² − (和)x + (积) = 0，即 x² + 3x − 10 = 0。

These formulas are derived directly from expanding (x − α)(x − β) = x² − (α + β)x + αβ. Once you see this, the formulas become much easier to remember.

这些公式直接来自展开 (x − α)(x − β) = x² − (α + β)x + αβ。一旦你理解了这一点，这些公式就更容易记住了。

In IGCSE papers, this topic sometimes appears alongside simultaneous equations: for example, you might be given that one root is twice the other, and asked to find the second coefficient. Set up the sum and product equations and solve them together.

在 IGCSE 试卷中，这个主题有时会与联立方程一起出现：例如，题目可能给出一个根是另一个根的两倍，然后要求你求另一个系数。建立和与积的方程组并一起求解。

8. Word Problems | 应用题

Quadratic equations frequently arise in geometry and number problems. The general strategy is to introduce a variable x, translate the given information into an equation, rearrange it into the standard form, and solve it using the most suitable method.

二次方程经常出现在几何和数字问题中。一般策略是引入变量 x，将给定信息转化为方程，将其整理为标准形式，然后用最合适的方法求解。

Example: A rectangle has length 4 cm more than its width. Its area is 45 cm². Find the width.

例题：一个矩形的长比宽多 4 cm。它的面积是 45 cm²。求宽。

Let the width be x cm. Then the length is (x + 4) cm. The area equation is x(x + 4) = 45, so x² + 4x − 45 = 0. Factorise: (x + 9)(x − 5) = 0, giving x = 5 or x = −9. Since a width cannot be negative, the answer is x = 5 cm. Always reject solutions that do not make sense in the context of the problem.

设宽为 x cm。则长为 (x + 4) cm。面积方程为 x(x + 4) = 45，即 x² + 4x − 45 = 0。因式分解：(x + 9)(x − 5) = 0，得到 x = 5 或 x = −9。由于宽度不能为负，答案是 x = 5 cm。始终舍弃在问题情境中不合理的解。

Other common contexts include projectile motion, where the height h(t) of an object is modelled by a quadratic in time t, and number puzzles such as ‘the product of two consecutive integers is 56’. In every case, check that your final answer is realistic and answer the question fully.

其他常见情境包括抛体运动（物体的高度 h(t) 是时间 t 的二次函数），以及数字谜题，如 “两个连续整数的积是 56″。在每种情况下，都要检查最终答案是否合理，并完整回答题目所问。

9. Common Mistakes | 常见错误

Nearly every mark lost on quadratic equation questions comes from one of a few repeated errors. Knowing these will help you avoid them in the exam.

二次方程题目中几乎每一分的丢失都源于几个反复出现的错误。了解这些错误将帮助你在考试中避免它们。

Error 1: Solving x² = 25 by writing x = 5 and forgetting the negative root. The correct answer is x = ±5. The ± sign matters whenever you take a square root.

错误 1：解 x² = 25 时写成 x = 5，忘记负根。正确答案是 x = ±5。每当你取平方根时，± 号都很重要。

Error 2: Cancelling x from both sides of an equation like x² = 3x. Dividing both sides by x loses the root x = 0. The correct approach is x² − 3x = 0, then factorise: x(x − 3) = 0, giving x = 0 or x = 3.

错误 2：在 x² = 3x 这样的方程两边同时消去 x。两边除以 x 会丢失根 x = 0。正确做法是 x² − 3x = 0，然后因式分解：x(x − 3) = 0，得到 x = 0 或 x = 3。

Error 3: Forgetting to rearrange the equation into standard form before applying the quadratic formula. If you substitute a, b, c from a randomised equation, the values will be wrong. Always write ax² + bx + c = 0 first.

错误 3：在使用二次公式前忘记将方程整理成标准形式。如果你从未整理的方程中读取 a、b、c 的值，结果就会出错。始终先写出 ax² + bx + c = 0。

Error 4: Sign errors when substituting negatives into the quadratic formula. Write every substitution in brackets, as shown in Section 4, and calculate carefully.

错误 4：将负数代入二次公式时出现符号错误。如第 4 节所示，将每个代入值写在括号内，并仔细计算。

10. Exam Tips and Practice | 考试技巧与练习

First, look at the equation before choosing a method. If the coefficients are small and factorisable, factorisation is fastest. If the equation contains awkward decimals or fractions, use the quadratic formula directly. If the question asks for the turning point, complete the square.

首先，在选择方法之前先观察方程。如果系数较小且可因式分解，因式分解最快。如果方程包含难处理的十进制数或分数，直接使用二次公式。如果题目要求顶点坐标，则使用配方法。

Always verify your answers by substituting them back into the original equation. This takes only a few seconds and catches most arithmetic errors. When the solution involves surds, you can leave your answer in exact form such as −3 ± √7; do not round unless the question asks for it.

始终通过将答案代回原方程来验证。这只需几秒钟，能发现大多数运算错误。当解涉及根式时，你可以将答案保留为精确形式，如 −3 ± √7；除非题目要求，否则不要四舍五入。

Try this quick practice pair:

试试这一组快速练习：

(a) Solve x² + 5x − 14 = 0. [Hint: factorise]

(a) 解 x² + 5x − 14 = 0。[提示：因式分解]

(b) Solve 2x² − 4x − 3 = 0, giving your answers correct to two decimal places. [Hint: use the formula]

(b) 解 2x² − 4x − 3 = 0，答案精确到小数点后两位。[提示：使用公式]

The answers are (a) x = 2 or x = −7, and (b) x ≈ 2.58 or x ≈ −0.58. If you got both correct, you are ready for the exam on this topic.

答案是 (a) x = 2 或 x = −7，(b) x ≈ 2.58 或 x ≈ −0.58。如果你都做对了，说明你已经准备好应对这个主题的考试了。

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Solving Quadratic Equations | 解二次方程