Quadratic Equations (Topic 234) | 二次方程（考点234）

📚 Quadratic Equations (Topic 234) | 二次方程（考点234）

Welcome to this focused revision guide on quadratic equations, specifically aligned with the Edexcel IGCSE Mathematics syllabus. This topic is essential for both Paper 1 and Paper 2, and mastering it will boost your confidence in algebra, graphing, and problem-solving.

欢迎阅读这本针对 Edexcel IGCSE 数学考试编写的二次方程复习指南。本主题在 Paper 1 和 Paper 2 中都是重点，掌握它能够提升你在代数、函数图象和实际问题求解方面的信心。

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

A quadratic equation is a polynomial equation of degree 2. Its standard form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The variable x is the unknown we aim to solve for.

二次方程是次数为 2 的多项式方程，其标准形式为 ax² + bx + c = 0，其中 a、b、c 均为常数，并且 a ≠ 0。变量 x 是我们要求解的未知数。

For example, 3x² − 5x + 2 = 0 is a quadratic equation because the highest power of x is 2. If a = 0, the equation becomes linear, so the term “quadratic” specifically requires a non-zero coefficient for x².

例如，3x² − 5x + 2 = 0 就是一个二次方程，因为 x 的最高次数为 2。如果 a = 0，方程就变成一次方程，因此“二次”这一名称要求 x² 的系数必须不为零。

Quadratic equations can have 0, 1, or 2 real solutions.
二次方程可能有 0 个、1 个或 2 个实数解。

2. Standard Form and Coefficients | 标准形式与系数

Before solving any quadratic, always rewrite it in the standard form ax² + bx + c = 0. This makes it easier to identify a, b, and c, which are used in factorisation, completing the square, and the quadratic formula.

在解任何二次方程之前，务必先将其改写为标准形式 ax² + bx + c = 0。这样可以更轻松地识别 a、b、c，它们在因式分解、配方法和二次公式中都会使用。

For example, the equation 2x² + 3x = 5 should be rearranged to 2x² + 3x − 5 = 0. Here a = 2, b = 3, and c = −5. Notice that the sign of c is negative, which affects the factorisation process.

例如，方程 2x² + 3x = 5 应重新排列为 2x² + 3x − 5 = 0。此时 a = 2、b = 3、c = −5。注意 c 的符号为负，这会影响因式分解的过程。

Equation	Standard Form	a	b	c
x² = 4x − 3	x² − 4x + 3 = 0	1	−4	3
2x(x − 1) = 0	2x² − 2x = 0	2	−2	0
−x² + 2 = 0	−x² + 0x + 2 = 0	−1	0	2

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer coefficients. The goal is to express ax² + bx + c as a product of two binomials, then set each factor equal to zero.

当二次方程的系数为简单整数时，因式分解通常是最快捷的方法。目标是将 ax² + bx + c 表示成两个二项式的乘积，然后令每个因子等于零。

Example 1: Solve x² + 5x + 6 = 0. We need two numbers that multiply to 6 and add to 5: these are 2 and 3. So (x + 2)(x + 3) = 0, giving x = −2 or x = −3.

例 1：解方程 x² + 5x + 6 = 0。我们需要找到两个数，它们的乘积为 6，和为 5：这两个数是 2 和 3。因此 (x + 2)(x + 3) = 0，得到 x = −2 或 x = −3。

Example 2: Solve 2x² − 3x − 2 = 0. We look for factors of 2 and −2 that produce the middle term −3x. The factorisation is (2x + 1)(x − 2) = 0, so x = −1/2 or x = 2.

例 2：解方程 2x² − 3x − 2 = 0。我们需要从 2 和 −2 的因子组合中，找到能产生中间项 −3x 的组合。因式分解为 (2x + 1)(x − 2) = 0，因此 x = −1/2 或 x = 2。

Always expand your factorisation to check that you have not made a sign error. If the quadratic cannot be factorised using integers, move on to another method.

一定要展开你的因式分解来检查是否出现符号错误。如果该二次方程无法用整数因式分解，就改用其他方法。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This method is useful for solving equations that do not factorise, and it also reveals the vertex of the parabola.

配方法将二次方程改写为 a(x + p)² + q 的形式。这种方法适用于无法因式分解的方程，同时也能揭示抛物线的顶点。

For a monic quadratic (a = 1), take half of the coefficient of x, square it, and adjust the constant term. For example, x² + 6x − 1 = 0 becomes (x + 3)² − 10 = 0.

对于首项系数为 1 的二次方程，取 x 系数的一半，平方后再调整常数项。例如，x² + 6x − 1 = 0 可转化为 (x + 3)² − 10 = 0。

Then solve: (x + 3)² = 10, so x + 3 = ±√10, giving x = −3 ± √10.

然后求解：(x + 3)² = 10，所以 x + 3 = ±√10，得到 x = −3 ± √10。

When a ≠ 1, first factor out a from the x terms. For instance, 2x² − 4x + 1 = 0 becomes 2[(x − 1)² − 1/2] = 0, simplifying to 2(x − 1)² − 1 = 0.

当 a ≠ 1 时，先从含 x 的项中提取公因数 a。例如，2x² − 4x + 1 = 0 可化为 2[(x − 1)² − 1/2] = 0，简化为 2(x − 1)² − 1 = 0。

5. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation, including those with irrational or complex roots. For ax² + bx + c = 0, the solutions are given by:

二次公式适用于所有二次方程，包括具有无理数根或复数根的方程。对于 ax² + bx + c = 0，其解为：

x = (−b ± √(b² − 4ac)) / 2a

This formula is derived from completing the square, and you are expected to memorise it for the Edexcel IGCSE exam. Make sure you substitute a, b, and c with their correct signs.

该公式由配方法推导而来，Edexcel IGCSE 考试中需要你记住它。代入 a、b、c 时一定要注意符号。

Worked example: Solve 3x² + 5x − 2 = 0. Here a = 3, b = 5, c = −2. The discriminant is 5² − 4(3)(−2) = 25 + 24 = 49. So x = (−5 ± √49) / 6 = (−5 ± 7) / 6. This gives x = 1/3 or x = −2.

例题：解方程 3x² + 5x − 2 = 0。此时 a = 3、b = 5、c = −2。判别式为 5² − 4(3)(−2) = 25 + 24 = 49。所以 x = (−5 ± √49) / 6 = (−5 ± 7) / 6，得到 x = 1/3 或 x = −2。

6. The Discriminant | 判别式

The discriminant, Δ = b² − 4ac, tells us the nature of the roots without solving the equation. It is a powerful tool for quick analysis.

判别式 Δ = b² − 4ac 可以让我们在不解方程的情况下判断根的性质。它是一个强大的快速分析工具。

If Δ > 0, the equation has two distinct real roots.
如果 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, the equation has exactly one repeated real root.
如果 Δ = 0，方程有一个重根（两个相等的实数根）。
If Δ < 0, the equation has no real roots (the roots are complex).
如果 Δ < 0，方程没有实数根（根为复数）。

For example, x² + 2x + 5 = 0 has Δ = 4 − 20 = −16, so it has no real solutions. Its graph will not cross the x-axis.

例如，x² + 2x + 5 = 0 的判别式为 Δ = 4 − 20 = −16，因此没有实数解。它的图象不会与 x 轴相交。

7. Roots and Coefficients (Vieta’s Formulas) | 根与系数的关系（韦达定理）

For the quadratic ax² + bx + c = 0 with roots α and β, the following relationships hold:

对于二次方程 ax² + bx + c = 0，若其根为 α 和 β，则成立以下关系：

α + β = −b/a and αβ = c/a

α + β = −b/a，且 αβ = c/a

These formulas are useful for finding the sum and product of roots without solving the equation. For example, for 2x² − 8x + 6 = 0, the sum of roots is 8/2 = 4 and the product is 6/2 = 3.

这些公式可以让我们在不解方程的情况下求出根的和与积。例如，对于 2x² − 8x + 6 = 0，根的和为 8/2 = 4，根的积为 6/2 = 3。

You may also be asked to form a new quadratic with given roots. If roots are p and q, the equation is x² − (p + q)x + pq = 0.

你还可能被要求根据给定的根来构造一个新二次方程。如果根为 p 和 q，则方程为 x² − (p + q)x + pq = 0。

8. Graphs of Quadratic Functions | 二次函数图象

The graph of y = ax² + bx + c is a parabola. If a > 0, it opens upward (a “U” shape); if a < 0, it opens downward (an "n" shape).

函数 y = ax² + bx + c 的图象是一条抛物线。如果 a > 0，开口向上（“U”形）；如果 a < 0，开口向下（“n”形）。

The x-coordinate of the vertex is given by x = −b/(2a). Substitute this back into the equation to find the y-coordinate. The intersection with the y-axis is at (0, c).

顶点的 x 坐标为 x = −b/(2a)。将其代回原方程可求得 y 坐标。与 y 轴的交点为 (0, c)。

The roots of the equation correspond to the x-intercepts of the graph. If the discriminant is negative, the graph does not touch the x-axis. For example, y = x² − 2x − 3 has vertex at (1, −4) and x-intercepts at x = −1 and x = 3.

方程的根对应图象与 x 轴的交点。如果判别式为负，则图象不与 x 轴相交。例如，y = x² − 2x − 3 的顶点为 (1, −4)，x 轴截距为 x = −1 和 x = 3。

9. Applications in Real Life | 实际应用

Quadratic equations appear in physics, economics, and geometry. For instance, when an object is thrown upward, its height h(t) after t seconds can be modelled by h(t) = −4.9t² + v₀t + h₀.

二次方程在物理、经济学和几何学中经常出现。例如，当物体竖直上抛时，t 秒后的高度 h(t) 可以用 h(t) = −4.9t² + v₀t + h₀ 来建模。

Solving such equations helps us find when the object reaches the ground (h = 0) or its maximum height. In business, profit functions are often quadratic, and finding the break-even points requires solving a quadratic equation.

求解这类方程可以帮助我们确定物体何时落地（h = 0）或达到最大高度。在商业中，利润函数常常是二次的，找出盈亏平衡点需要解二次方程。

Another example is the area of a rectangular field: if one side is x metres and the perimeter is fixed, the area is given by a quadratic expression. Maximising this area involves finding the vertex of the parabola.

另一个例子是矩形场地的面积：如果一边长为 x 米，且周长固定，则面积为二次表达式。求面积最大值需要找到抛物线的顶点。

10. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Below are frequent errors students make when working with quadratics, along with tips to fix them.

下面是学生在处理二次方程时常犯的错误，以及纠正这些错误的方法。

Forgetting to set the equation to zero before factorising. Always rearrange to ax² + bx + c = 0 first.
忘记在因式分解前将方程化为零。务必先整理成 ax² + bx + c = 0。
Misidentifying the sign of c when a is negative. For example, −x² + 3x − 2 = 0 should be treated with a = −1, b = 3, c = −2.
当 a 为负数时弄错 c 的符号。例如，−x² + 3x − 2 = 0 应被视为 a = −1、b = 3、c = −2。
Using the quadratic formula with a = 0, which is undefined. Ensure the equation is truly quadratic.
在 a = 0 时使用二次公式，这是无定义的。要确保方程确实是二次方程。
Losing solutions when taking square roots. Remember that x² = k implies x = ±√k.
在求平方根时丢失解。记住 x² = k 意味着 x = ±√k。

11. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧

In the exam, read the question carefully. If it asks for “exact values”, leave your answer in surd form or as a fraction, not a rounded decimal. If it asks for “correct to 2 decimal places”, use your calculator accurately.

考试中要仔细阅读题目。如果要求“精确值”，请将答案保留为根式或分数，不要四舍五入成小数。如果要求“精确到小数点后两位”，请准确使用计算器。

Practice solving quadratics by all three methods: factorisation, completing the square, and the quadratic formula. Be ready to choose the most efficient method based on the equation.

练习用三种方法解二次方程：因式分解法、配方法和二次公式。要根据方程的特点选择最有效的方法。

Always check your answers by substituting them back into the original equation. This catches simple arithmetic errors. Also, sketch a quick graph when dealing with inequalities or range of values.

一定要将答案代回原方程进行检验，这能发现简单的算术错误。处理不等式或取值区间时，快速画一个草图也会很有帮助。

Finally, manage your time. Quadratic questions can appear as multiple-choice, short answer, or problem solving. Aim to recognise the type of question and apply the relevant technique confidently.

最后，合理分配时间。二次方程可能以选择题、简答题或应用题的形式出现。要能够识别题型并自信地运用相应技巧。

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Quadratic Equations (Topic 234) | 二次方程（考点234）