Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear throughout the Edexcel IGCSE Mathematics syllabus. They are central to algebra, graphs, and problem solving. This guide explains every method you need, working through each technique step by step so you can approach any quadratic question with confidence.

二次方程贯穿 Edexcel IGCSE 数学的整个考纲，是代数、函数图像和实际应用题的核心。本文将逐一讲解所有解题方法，手把手带你练透每种技巧，让你面对任何二次方程题型都能从容应对。

1. What Is a Quadratic Equation? | 什么是一元二次方程？

A quadratic equation is any equation that can be written in the standard form:

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants, and the highest power of x is 2. The value of a must not be zero; if a = 0, the equation becomes linear. For example, 3x² − 5x + 2 = 0 is quadratic, but 2x − 1 = 0 is not.

其中 a、b、c 为常数，x 的最高次数是 2。a 一定不能为零，否则方程就退化成了一次方程。例如 3x² − 5x + 2 = 0 是二次方程，而 2x − 1 = 0 不是。

Every quadratic equation has at most two real solutions, which are also called roots. Some have two distinct roots, some have exactly one repeated root, and some have no real roots at all. The next sections show you how to find them.

每个二次方程最多有两个实数解，也叫做根。有的有两个不同的根，有的恰好有一个重根，还有的完全没有实数根。接下来的章节将教你如何求根。

2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method, but it only works when the roots are rational numbers. The rule you use is called the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

因式分解是最快的方法，但只适用于根为有理数的情况。其核心依据是”零乘积性质”：若两个表达式相乘等于零，则其中至少有一个为零。

For an equation in the form x² + bx + c = 0, look for two numbers that multiply to give c and add to give b. For example:

对于 x² + bx + c = 0 这种形式的方程，找两个数，使它们的乘积等于 c，和等于 b。例如：

x² − 7x + 12 = 0

The numbers are −3 and −4 because (−3) × (−4) = 12 and (−3) + (−4) = −7. So the equation becomes (x − 3)(x − 4) = 0, giving x = 3 or x = 4.

这两个数是 −3 和 −4，因为 (−3) × (−4) = 12 且 (−3) + (−4) = −7。于是原方程化为 (x − 3)(x − 4) = 0，解得 x = 3 或 x = 4。

When the coefficient of x² is not 1, use the ac method or factor by grouping. Try 2x² + 7x + 3 = 0. Here a × c = 6, and two factors of 6 that add to 7 are 1 and 6. Rewrite the middle term:

当 x² 的系数不为 1 时，可采用 ac 法或分组分解。以 2x² + 7x + 3 = 0 为例，a × c = 6，6 的两个因数中相加得 7 的是 1 和 6。改写中间项：

2x² + x + 6x + 3 = 0

Then group the terms: x(2x + 1) + 3(2x + 1) = 0, so (2x + 1)(x + 3) = 0. The solutions are x = −1/2 and x = −3. Always check whether the original equation can be factorised before reaching for a longer method.

然后分组：x(2x + 1) + 3(2x + 1) = 0，即 (2x + 1)(x + 3) = 0，解得 x = −1/2 和 x = −3。做题时先判断能否因式分解，再决定是否使用更复杂的方法。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. This method works for every quadratic equation and is especially useful for finding turning points, as you will see later.

配方法将二次式改写为 (x + p)² + q 的形式。这种方法对所有二次方程都适用，尤其在求顶点坐标时非常有用，后文会具体说明。

Start with x² + 6x − 7 = 0. Take half of 6, which is 3, and write (x + 3)². Since (x + 3)² = x² + 6x + 9, you must subtract 9 to keep the expression equal:

从 x² + 6x − 7 = 0 开始。取 6 的一半，即 3，写成 (x + 3)²。因为 (x + 3)² = x² + 6x + 9，所以必须减去 9 才能保持等式成立：

(x + 3)² − 9 − 7 = 0, 即 (x + 3)² = 16

Now take square roots on both sides: x + 3 = ±4. Therefore x = 1 or x = −7. Remember the ± symbol: when you square root, you must consider both the positive and negative cases.

两边同时开平方：x + 3 = ±4，因此 x = 1 或 x = −7。记得写 ± 符号：开平方时必须同时考虑正负两种情况。

If the coefficient of x² is not 1, factor it out first. For 2x² + 8x + 5 = 0, write 2(x² + 4x) + 5 = 0, then complete the square inside the bracket: 2[(x + 2)² − 4] + 5 = 0, which simplifies to 2(x + 2)² − 3 = 0.

如果 x² 的系数不是 1，先将其提取出来。对于 2x² + 8x + 5 = 0，写成 2(x² + 4x) + 5 = 0，再在括号内配方：2[(x + 2)² − 4] + 5 = 0，化简得 2(x + 2)² − 3 = 0。

4. The Quadratic Formula | 求根公式

The quadratic formula is a universal tool. It works for every quadratic equation, including those that cannot be factorised easily. The formula is:

求根公式是解决一切二次方程的万能工具，即使方程难以因式分解，它也能轻松应对。公式如下：

x = (−b ± √(b² − 4ac)) ÷ 2a

To use it, identify a, b and c from ax² + bx + c = 0, then substitute. For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4 and c = −3. Substituting:

使用时先从 ax² + bx + c = 0 中找出 a、b、c，然后代入。例如解 2x² − 4x − 3 = 0，其中 a = 2，b = −4，c = −3。代入得：

x = (4 ± √((−4)² − 4 × 2 × (−3))) ÷ (2 × 2)

This becomes x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4. Since √40 = 2√10, the final answer is x = (2 ± √10) ÷ 2. In exam questions, you may be asked to leave answers in exact surd form rather than as decimals.

整理得 x = (4 ± √(16 + 24)) ÷ 4 = (4 ± √40) ÷ 4。由于 √40 = 2√10，最终答案为 x = (2 ± √10) ÷ 2。考试中常要求保留根号的精确形式，而不是写成小数。

5. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant, often written as Δ. Its value tells you the number and type of roots without solving the whole equation.

根号内的表达式 b² − 4ac 叫作判别式，通常记作 Δ。通过它的值，不需要完全解方程就能判断根的个数和类型。

Δ > 0: two distinct real roots (the graph crosses the x-axis twice).
Δ = 0: one repeated real root (the graph touches the x-axis once).
Δ < 0: no real roots (the graph does not touch the x-axis).

Δ > 0：两个不同的实数根（图像与 x 轴有两个交点）。
Δ = 0：一个重根（图像与 x 轴相切于一点）。
Δ < 0：没有实数根（图像与 x 轴无交点）。

For example, consider x² + 4x + 5 = 0. The discriminant is 4² − 4 × 1 × 5 = 16 − 20 = −4, which is negative, so there are no real roots. This kind of reasoning is often tested in multiple-choice questions, so practise identifying Δ quickly.

例如，x² + 4x + 5 = 0 的判别式为 4² − 4 × 1 × 5 = 16 − 20 = −4，是负数，因此没有实数根。这类判断常出现在选择题中，务必练习快速计算 Δ。

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

A quadratic function y = ax² + bx + c always produces a parabola. The sign of a determines the direction of the curve: if a > 0, the parabola opens upwards like a U shape; if a < 0, it opens downwards like an inverted U.

二次函数 y = ax² + bx + c 的图像永远是抛物线。a 的符号决定开口方向：当 a > 0 时开口向上，呈 U 形；当 a < 0 时开口向下，呈倒 U 形。

The graph provides a visual interpretation of the roots: the x-intercepts are the roots of the equation ax² + bx + c = 0. The y-intercept is simply c, because when x = 0, y = c. For example, y = x² − 4x + 3 has roots x = 1 and x = 3, and it crosses the y-axis at (0, 3).

图像直观地展示了根的含义：抛物线与 x 轴的交点就是方程 ax² + bx + c = 0 的根。与 y 轴的交点就是 c，因为当 x = 0 时 y = c。例如 y = x² − 4x + 3 的根为 x = 1 和 x = 3，且与 y 轴交于 (0, 3)。

When sketching a quadratic graph, always label the roots, the y-intercept, and the turning point. These three features are enough to draw an accurate parabola. If the equation has no real roots, note that the curve does not cross the x-axis at all.

画二次函数草图时，一定要标出根、y 轴截距和顶点。有了这三个关键点，就能画出准确的抛物线。如果方程没有实数根，曲线就完全不与 x 轴相交。

7. Finding Turning Points | 求顶点坐标

The turning point of a parabola is its minimum (when a > 0) or maximum (when a < 0). There are two reliable ways to find it. The first method uses completing the square: if y = (x + p)² + q, then the vertex is at (−p, q).

抛物线的顶点就是它的最小值（当 a > 0 时）或最大值（当 a < 0 时）。求顶点有两种可靠方法。第一种是用配方法：若 y = (x + p)² + q，则顶点坐标为 (−p, q)。

Take y = x² − 6x + 5. Complete the square: y = (x − 3)² − 9 + 5 = (x − 3)² − 4. Therefore the vertex is at (3, −4), and since a = 1 > 0, this is a minimum point.

以 y = x² − 6x + 5 为例，配方得 y = (x − 3)² − 9 + 5 = (x − 3)² − 4，因此顶点为 (3, −4)。因为 a = 1 > 0，所以这是一个最小值点。

The second method uses symmetry. The axis of symmetry of the parabola is the vertical line x = −b ÷ (2a). For y = x² − 6x + 5, this gives x = 6 ÷ 2 = 3, the same x-coordinate of the vertex. Substitute into the equation to find the y-coordinate.

第二种方法利用对称性。抛物线的对称轴是竖直线 x = −b ÷ (2a)。对于 y = x² − 6x + 5，可得 x = 6 ÷ 2 = 3，与顶点的 x 坐标相同。再代入原式求出 y 坐标。

8. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities involve expressions like ax² + bx + c > 0 or ax² + bx + c < 0. The key is to solve the corresponding equation first to find the boundary points, then use the shape of the graph to determine the solution region.

二次不等式形如 ax² + bx + c > 0 或 ax² + bx + c < 0。解题关键是先解对应的方程，找到边界点，再利用抛物线开口方向来确定解集区域。

Consider x² − 4x − 5 < 0. First factorise: (x − 5)(x + 1) = 0, so the boundary points are x = 5 and x = −1. Since a = 1 > 0, the parabola opens upwards, which means the curve is below the x-axis between the roots. Therefore the solution is −1 < x < 5.

以 x² − 4x − 5 < 0 为例。先因式分解：(x − 5)(x + 1) = 0，边界点为 x = 5 和 x = −1。因为 a = 1 > 0，抛物线开口向上，曲线在两个根之间位于 x 轴下方，所以解集为 −1 < x < 5。

If the inequality sign is reversed, x² − 4x − 5 > 0, the solution is the outer region: x < −1 or x > 5. Sketching a quick graph is the safest way to avoid sign errors, especially when the coefficient of x² is negative.

如果不等号方向相反，即 x² − 4x − 5 > 0，则解集为外部区域：x < −1 或 x > 5。快速画草图是避免符号错误最稳妥的方法，尤其是当 x² 的系数为负数时。

9. Applications in Problem Solving | 实际应用

Quadratic equations often appear in word problems involving areas, motion, or consecutive numbers. The standard strategy is to translate the sentence into an equation, simplify it to the form ax² + bx + c = 0, and then solve.

二次方程经常出现在与面积、运动或连续整数相关的应用题中。标准策略是把文字描述转化为方程，化简为 ax² + bx + c = 0 的形式，然后再求解。

For example, a rectangle has length (x + 3) cm and width x cm, with an area of 28 cm². Since area = length × width, we get x(x + 3) = 28, so x² + 3x − 28 = 0. Factorising gives (x + 7)(x − 4) = 0, so x = −7 or x = 4. Since length cannot be negative, x = 4, and the dimensions are 4 cm by 7 cm.

例如，一个长方形的长为 (x + 3) cm，宽为 x cm，面积为 28 cm²。由面积 = 长 × 宽，得 x(x + 3) = 28，即 x² + 3x − 28 = 0。因式分解得 (x + 7)(x − 4) = 0，所以 x = −7 或 x = 4。由于长度不能为负，取 x = 4，即宽 4 cm、长 7 cm。

Always check whether your answers make sense in context. A negative root is often algebraically valid but physically impossible. Also remember to state the final answer with the correct units.

务必检查答案在实际情境中是否合理。负根在代数上往往成立，但在实际中不可能。同时，记得在最终答案中带上正确的单位。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

Here are the most common errors students make with quadratics, along with exam strategies to avoid them. Practising these points will save you marks on exam day.

以下是最常见的二次方程错误以及对应的考试应对策略。在平时练习中注意这些要点，考试时能帮你少丢分。

Rearrange to standard form first. Always move every term to one side so the equation equals zero before solving. For example, x² = 5x − 6 must become x² − 5x + 6 = 0.
Do not forget the ± sign. When taking square roots in completing the square, always write ±. Missing it loses one solution.
Check the discriminant before using the formula. If Δ < 0, stop and state that there are no real roots; do not waste time substituting.
Handle negative b carefully. In the formula, if b = −4, then −b is +4. A common slip is keeping the minus sign.
Verify by substitution. After solving, plug your answers back into the original equation to confirm they work.

先整理成标准形式。解题前先把所有项移到同一边，使方程等于 0。例如 x² = 5x − 6 必须改写为 x² − 5x + 6 = 0。
不要漏掉 ± 符号。配方法开平方时，务必写 ±，否则会丢掉一个解。
用公式前先看判别式。若 Δ < 0，直接写"无实数根"，不要浪费时间代入。
小心处理负 b。公式中若 b = −4，则 −b = +4，常见错误是保留负号。
代入验算。解完之后把答案代回原方程，确认是否成立。

By mastering the five key tools — factorisation, completing the square, the quadratic formula, the discriminant, and graph sketching — you will be fully prepared for any quadratic question on the Edexcel IGCSE Mathematics paper.

掌握这五大工具——因式分解、配方法、求根公式、判别式和图像作图——你就能从容应对 Edexcel IGCSE 数学试卷中的任何二次方程题目。

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Solving Quadratic Equations | 解一元二次方程