📚 Solving Quadratic Equations | 二次方程求解方法
Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. Mastering the various methods of solving them is essential for both Paper 2 (non-calculator) and Paper 4 (calculator) examinations. This article provides a comprehensive review of the key techniques, worked examples, and common pitfalls to avoid.
二次方程是 IGCSE 数学中考查频率最高的知识点之一。掌握其各种求解方法，对于 Paper 2（非计算器）和 Paper 4（计算器）考试都至关重要。本文将系统梳理核心解法、典型例题及常见易错点，帮助你稳拿高分。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants, with a ≠ 0. The value of a must not be zero; otherwise, the equation becomes linear. The terms ax², bx and c are called the quadratic term, linear term and constant term respectively.
二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。a 的值不能为零，否则方程就变成了线性方程。ax²、bx、c 三项分别称为二次项、一次项和常数项。
ax² + bx + c = 0 (a ≠ 0)
For example, 2x² − 5x + 3 = 0 and x² − 9 = 0 are both quadratic equations, while x³ − 2x + 1 = 0 is not. In many exam questions, you will first need to rearrange the given equation into this standard form before applying any solving technique.
例如，2x² − 5x + 3 = 0 和 x² − 9 = 0 都是二次方程，而 x³ − 2x + 1 = 0 则不是。在许多考试题目中，你需要先将所给方程整理成标准形式，然后再应用求解方法。
2. Solving by Factorisation | 因式分解法
Factorisation is the most direct method when the quadratic can be expressed as a product of two linear factors. The principle is based on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.
因式分解法是当二次式能写成两个一次因式乘积时的最直接方法。其原理基于零积性质：如果两个表达式的乘积为零，那么其中至少有一个必须为零。
To solve x² + bx + c = 0 by factorisation, look for two numbers whose sum is b and whose product is c. For the general form ax² + bx + c = 0 where a ≠ 1, you may need to use decomposition or trial and error.
用因式分解法解 x² + bx + c = 0 时，寻找两个数，使它们的和为 b、积为 c。对于一般形式 ax² + bx + c = 0（a ≠ 1），则可能需要使用十字相乘法或试错法。
Worked Example | 例题： Solve x² − 7x + 12 = 0.
Find two numbers with sum 7 and product 12: 3 and 4. Thus x² − 7x + 12 = (x − 3)(x − 4) = 0, giving x = 3 or x = 4.
寻找和为 7、积为 12 的两个数：3 和 4。于是 x² − 7x + 12 = (x − 3)(x − 4) = 0，得到 x = 3 或 x = 4。
3. Solving by Completing the Square | 配方法
Completing the square transforms ax² + bx + c = 0 into the form a(x + p)² + q = 0. This method is particularly useful when the equation cannot be factorised easily, and it also reveals the coordinates of the vertex of the corresponding parabola.
配方法将 ax² + bx + c = 0 转化为 a(x + p)² + q = 0 的形式。当方程不易因式分解时，此方法尤为实用，同时还能揭示对应抛物线的顶点坐标。
The general procedure: divide by a if necessary, then halve the coefficient of x, square it, and add/subtract accordingly to maintain equivalence.
一般步骤：必要时先除以 a，然后将 x 的系数取半、平方，并相应加减以保持等式成立。
Worked Example | 例题： Solve x² + 6x − 7 = 0 by completing the square.
Halve 6 to get 3, square to get 9. Rewrite: x² + 6x = (x + 3)² − 9. Thus the equation becomes (x + 3)² − 9 − 7 = 0 → (x + 3)² = 16 → x + 3 = ±4 → x = 1 or x = −7.
将 6 取半得 3，平方得 9。改写：x² + 6x = (x + 3)² − 9。于是方程变为 (x + 3)² − 9 − 7 = 0 → (x + 3)² = 16 → x + 3 = ±4 → x = 1 或 x = −7。
4. The Quadratic Formula | 求根公式
The quadratic formula is a universal method that solves any quadratic equation, whether or not it can be factorised. For ax² + bx + c = 0, the solutions are given by the formula below. You are expected to memorise this formula for the IGCSE examination.
求根公式是一种通用方法，可解任何二次方程，无论其能否因式分解。对于 ax² + bx + c = 0，解由下列公式给出。IGCSE 考试要求记住此公式。
x = (−b ± √(b² − 4ac)) / (2a)
The expression b² − 4ac under the square root is called the discriminant. Its value determines the nature of the roots, as discussed in the next section.
根号内的表达式 b² − 4ac 称为判别式。它的值决定了根的性质，将在下一节详细讨论。
Worked Example | 例题： Solve 2x² + 3x − 5 = 0 using the formula.
Here a = 2, b = 3, c = −5. Substitute: x = (−3 ± √(9 + 40)) / 4 = (−3 ± 7) / 4. Hence x = 1 or x = −2.5.
这里 a = 2、b = 3、c = −5。代入得：x = (−3 ± √(9 + 40)) / 4 = (−3 ± 7) / 4。因此 x = 1 或 x = −2.5。
5. The Discriminant | 判别式
The discriminant, denoted by Δ (or sometimes just written as b² − 4ac), tells us about the number and type of roots without actually solving the equation. This is a frequent topic in IGCSE multiple-choice and short-answer questions.
判别式（记作 Δ，有时直接写作 b² − 4ac）可以让我们无需解方程就能判断根的个数和类型。这是 IGCSE 选择和简答题中的常见考点。
| Discriminant Value | 判别式值 | Nature of Roots | 根的性质 | Graphical Meaning | 图像意义 |
|---|---|---|
| Δ > 0 | Two distinct real roots 两个不等实根 |
Parabola crosses the x-axis at two points 抛物线与 x 轴有两个交点 |
| Δ = 0 | One repeated real root 两个相等实根（一个重根） |
Parabola touches the x-axis at one point 抛物线与 x 轴相切于一点 |
| Δ < 0 | No real roots 无实根 |
Parabola does not intersect the x-axis 抛物线与 x 轴无交点 |
When a question asks you to “find the range of values of k for which the equation has two distinct real roots,” you would set Δ > 0 and solve the resulting inequality.
当题目要求“求 k 的取值范围，使方程有两个不等实根”时，你需要令 Δ > 0 并解所得不等式。
6. Solving Quadratic Inequalities | 二次不等式
Quadratic inequalities extend the idea of solving equations to finding intervals of x that satisfy an inequality. The most reliable approach is to first solve the corresponding equation, sketch the parabola, and then read off the required intervals.
二次不等式将解方程的思想扩展到寻找满足不等式的 x 的区间。最可靠的方法是：先解对应方程，画出抛物线草图，然后读出所需区间。
Worked Example | 例题： Solve x² − 5x + 6 > 0.
First factorise: (x − 2)(x − 3) > 0. The critical points are x = 2 and x = 3. The parabola opens upwards, so the expression is positive when x < 2 or x > 3.
首先因式分解：(x − 2)(x − 3) > 0。临界点为 x = 2 和 x = 3。抛物线开口向上，因此当 x < 2 或 x > 3 时，表达式为正。
x < 2 or x > 3
For a “less than” inequality such as x² − 5x + 6 < 0, the solution would instead be 2 < x < 3. Remember that the inequality sign flips only when multiplying or dividing by a negative number, which is not the case in pure factorisation.
对于“小于”不等式，如 x² − 5x + 6 < 0，解则为 2 < x < 3。注意：只有在乘以或除以负数时不等号才需要变向，纯因式分解过程中不需要。
7. Word Problems Involving Quadratics | 二次方程应用题
IGCSE examination papers frequently include word problems that require you to form a quadratic equation from a real-life context and then solve it. Common scenarios include area problems, number problems and projectile motion.
IGCSE 考卷中常有需要你从实际情境中建立二次方程并求解的应用题。常见情境包括面积问题、数字问题和抛体运动问题。
Worked Example | 例题： The length of a rectangle is 4 cm more than its width. Its area is 45 cm². Find the width.
例题：矩形的长比宽多 4 cm，面积为 45 cm²。求宽。
Let the width be x. Then the length is x + 4, and x(x + 4) = 45 → x² + 4x − 45 = 0 → (x + 9)(x − 5) = 0 → x = 5 (reject x = −9 since length cannot be negative).
设宽为 x，则长为 x + 4，且 x(x + 4) = 45 → x² + 4x − 45 = 0 → (x + 9)(x − 5) = 0 → x = 5（舍去 x = −9，因为长度不能为负）。
Always check whether extraneous solutions should be rejected based on the context. Also remember to state your final answer with the correct unit.
务必根据实际情境判断是否需要舍去增根。同时记得在最终答案中带上正确单位。
8. Graphs of Quadratic Functions | 二次函数图像
The graph of y = ax² + bx + c is a parabola. The sign of a determines the direction of the opening: upwards when a > 0, downwards when a < 0. The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola.
函数 y = ax² + bx + c 的图像是一条抛物线。a 的符号决定开口方向：a > 0 时开口向上，a < 0 时开口向下。方程 ax² + bx + c = 0 的根对应抛物线与 x 轴的交点横坐标。
Axis of symmetry: x = −b / (2a). Vertex (turning point): substitute this x-value into the equation to find y.
对称轴：x = −b / (2a)。顶点（转折点）：将此 x 值代入方程求出 y。
For example, for y = x² − 6x + 5, the axis of symmetry is x = 3, and the vertex is (3, −4). The roots are x = 1 and x = 5, so the graph crosses the x-axis at those points.
例如，对于 y = x² − 6x + 5，对称轴为 x = 3，顶点为 (3, −4)。根为 x = 1 和 x = 5，因此图像在这两点与 x 轴相交。
When sketching a quadratic graph, always clearly label: the roots, the y-intercept, the vertex, and the axis of symmetry.
画二次函数图像草图时，务必清晰标注：根、y 轴截距、顶点和对称轴。
9. Sum and Product of Roots | 根与系数的关系
For the quadratic equation ax² + bx + c = 0 with roots α and β, the following relationships hold. These are useful shortcuts for certain exam questions but are not always explicitly listed in the formula booklet.
对于根为 α 和 β 的二次方程 ax² + bx + c = 0，有以下关系成立。这些是解某些考题时的快捷工具，但未必在公式册中直接列出。
Sum of roots: α + β = −b/a
Product of roots: αβ = c/a
For example, if a quadratic equation has roots 3 and −2, then the sum is 1 and the product is −6, so a possible equation is x² − x − 6 = 0.
例如，若一个二次方程的根为 3 和 −2，则两和根为 1、两根积为 −6，因此一个可能的方程为 x² − x − 6 = 0。
This technique is especially handy when a question gives the roots and asks you to reconstruct the equation, without requiring you to expand brackets unnecessarily.
当题目给出根并要求你还原方程时，此法尤其便捷，不必多余地展开括号。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common Mistake 1 | 常见错误一： Forgetting to rearrange the equation into standard form before applying the quadratic formula. Always write ax² + bx + c = 0 first.
常见错误一：在使用求根公式前忘记将方程整理成标准形式。务必先写出 ax² + bx + c = 0。
Common Mistake 2 | 常见错误二： Sign errors when substituting negative values of b or c into the quadratic formula. Use brackets carefully in your calculator.
常见错误二：将 b 或 c 的负值代入求根公式时出现符号错误。在计算器中谨慎使用括号。
Common Mistake 3 | 常见错误三： Dividing both sides of an equation by a term containing the variable x, which may cause loss of a root. Only divide by a non-zero constant.
常见错误三：等式两边除以含有变量 x 的项，导致失去一个根。只能除以非零常数。
Exam tip: when using the quadratic formula on a non-calculator paper, leave your answer in surd form rather than converting to a decimal unless the question specifies otherwise.
考试技巧：在非计算器试卷上使用求根公式时，答案保留根式形式，除非题目明确要求保留小数。
Remember to always check your solutions by substituting back into the original equation. This simple step can prevent losing marks to careless arithmetic errors.
切记将解代回原方程进行验算。这一简单步骤可以防止因粗心计算错误而失分。
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