Mastering Quadratic Equations | 掌握二次方程

📚 Mastering Quadratic Equations | 掌握二次方程

Quadratic equations are one of the most essential topics in the IGCSE Mathematics syllabus. They appear in nearly every exam paper, whether as standalone questions or as part of problem-solving scenarios. Understanding how to solve and analyse quadratic equations is not just about memorising formulas; it is about developing a flexible mathematical mindset.

二次方程是 IGCSE 数学大纲中最核心的主题之一。几乎每份试卷都会出现二次方程，无论是独立题目还是作为应用题的一部分。掌握二次方程的解法与分析，不仅是记忆公式，更是培养灵活的数学思维。

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 highest power of the variable x is 2, which is why it is called ‘quadratic’ — from the Latin word ‘quadratus’, meaning square.

二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数，且 a ≠ 0。变量 x 的最高次数为 2，因此被称为 ‘quadratic’ —— 源自拉丁语 ‘quadratus’，意为平方。

ax² + bx + c = 0

For example, 2x² – 5x + 3 = 0 is a quadratic equation, while x² – 4x + 2 = 0 is another classic example. When a = 0, the equation degenerates into a linear equation, which is why the condition a ≠ 0 is crucial.

例如，2x² – 5x + 3 = 0 是二次方程，x² – 4x + 2 = 0 也是经典例子。当 a = 0 时，方程退化为一次线性方程，因此 a ≠ 0 这一条件至关重要。

2. Solving by Factorisation | 因式分解法

Factorisation is often the first method taught for solving quadratic equations. The key idea is to rewrite ax² + bx + c as a product of two linear expressions. For simple cases where a = 1, we look for two numbers whose product is c and whose sum is b.

因式分解法通常是学习解二次方程的第一种方法。核心思想是将 ax² + bx + c 重写为两个一次表达式的乘积。对于 a = 1 的简单情况，我们寻找两个数，使其乘积为 c，和为 b。

Consider the equation x² + 5x + 6 = 0. We need two numbers that multiply to 6 and add to 5. These numbers are 2 and 3. Therefore, the equation factors as (x + 2)(x + 3) = 0. Setting each factor to zero gives x = -2 or x = -3.

考虑方程 x² + 5x + 6 = 0。我们需要找到两个数，相乘得 6，相加得 5，这两个数是 2 和 3。因此，方程分解为 (x + 2)(x + 3) = 0。令每个因式等于零，得到 x = -2 或 x = -3。

If (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0

When a ≠ 1, the process requires more care. For instance, to factor 2x² + 7x + 3, we look for factors of 2 × 3 = 6 that sum to 7. The numbers 6 and 1 work, allowing us to split the middle term: 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Thus x = -1/2 or x = -3.

当 a ≠ 1 时，因式分解需要更多技巧。例如，分解 2x² + 7x + 3 时，我们寻找 2 × 3 = 6 的因数，且其和为 7。数字 6 和 1 满足条件，于是我们使用拆项法：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。因此 x = -1/2 或 x = -3。

3. Solving by Completing the Square | 配方法

Completing the square is a powerful algebraic technique that transforms any quadratic equation into a perfect square plus a constant. This method not only solves equations but also reveals the vertex of the corresponding parabola.

配方法是一种强有力的代数技巧，它可以将任意二次方程转化为一个完全平方加常数的形式。这种方法不仅能解方程，还能揭示对应抛物线的顶点。

Starting with x² + bx, we add and subtract (b/2)² to create a perfect square. For the equation x² + 6x – 7 = 0, we first move the constant: x² + 6x = 7. Then we complete the square on the left: x² + 6x + 9 = 16, which gives (x + 3)² = 16. Taking square roots yields x + 3 = ±4, so x = 1 or x = -7.

从 x² + bx 开始，我们加减 (b/2)² 来构造完全平方。对于方程 x² + 6x – 7 = 0，先移常数项：x² + 6x = 7。然后在左边配方：x² + 6x + 9 = 16，即 (x + 3)² = 16。两边开平方得到 x + 3 = ±4，所以 x = 1 或 x = -7。

(x + b/2)² = (b/2)² – c

In general, for x² + bx + c = 0, the solutions are found by writing (x + b/2)² = (b/2)² – c. This method is especially useful when the quadratic does not factor nicely, or when we need to determine the maximum or minimum value of a quadratic function.

一般而言，对于 x² + bx + c = 0，可以通过 (x + b/2)² = (b/2)² – c 求出解。当二次方程不易因式分解，或需要确定二次函数的最大值或最小值时，这种方法尤为有用。

4. The Quadratic Formula | 求根公式

The quadratic formula is the most general and reliable method for solving any quadratic equation. Derived from completing the square, it provides solutions directly in terms of the coefficients a, b and c.

求根公式是解任意二次方程最通用、最可靠的方法。它由配方法推导而来，直接以系数 a、b、c 给出解。

x = (-b ± √(b² – 4ac)) / 2a

To use the formula, simply substitute the values of a, b and c into the equation and simplify. For example, solving 2x² – 4x – 3 = 0, we have a = 2, b = -4, c = -3. Substituting gives x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± (√10)/2.

使用公式时，只需将 a、b、c 的值代入并化简。例如，解 2x² – 4x – 3 = 0 时，a = 2，b = -4，c = -3。代入得 x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± (√10)/2。

The quadratic formula always works, even when factorisation is impossible. It is essential to simplify square roots fully and to check whether the final answer can be written in a simpler form.

求根公式永远有效，即使无法因式分解也能求解。务必完整化简平方根，并检查最终答案是否可以写成更简单的形式。

5. The Discriminant and Nature of Roots | 判别式与根的性质

The expression b² – 4ac is called the discriminant, usually denoted by the Greek letter Δ. It determines the nature of the roots without actually solving the equation.

表达式 b² – 4ac 称为判别式，通常用希腊字母 Δ 表示。它无需解方程就能确定根的性质。

If Δ > 0, the equation has two distinct real roots. The parabola crosses the x-axis at two different points.

若 Δ > 0，方程有两个不相等的实数根，抛物线在 x 轴上有两个交点。
If Δ = 0, the equation has exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point, known as the vertex.

若 Δ = 0，方程有且仅有一个实数根（重根），抛物线在 x 轴上恰好有一个切点，即顶点。
If Δ < 0, the equation has no real roots. The parabola does not intersect the x-axis at all.

若 Δ < 0，方程没有实数根，抛物线与 x 轴没有交点。

For instance, the equation x² – 4x + 4 = 0 has Δ = 16 – 16 = 0, so it has one repeated root x = 2. The equation x² + x + 1 = 0 has Δ = 1 – 4 = -3, indicating no real roots.

例如，方程 x² – 4x + 4 = 0 的 Δ = 16 – 16 = 0，因此有一个重根 x = 2。方程 x² + x + 1 = 0 的 Δ = 1 – 4 = -3，表明没有实数根。

6. The Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, there are elegant relationships between the roots and the coefficients. These formulas allow us to derive information about the roots without solving the equation.

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根与系数之间存在优美的关系。这些公式让我们无需解方程就能获得关于根的信息。

Sum of roots: α + β = -b/a

Product of roots: α × β = c/a

These relationships are particularly useful for forming new quadratic equations with prescribed roots. If we want a quadratic with roots α and β, we can write x² – (α + β)x + αβ = 0.

这些关系在构建具有指定根的二次方程时特别有用。如果我们需要一个根为 α 和 β 的二次方程，可以写成 x² – (α + β)x + αβ = 0。

For example, if the roots of a quadratic are 3 and -2, then the sum is 1 and the product is -6, giving x² – x – 6 = 0. This approach is frequently tested in IGCSE papers and is a quick way to verify factorisation results.

例如，若某二次方程的根为 3 和 -2，则根的和为 1，积为 -6，因此方程为 x² – x – 6 = 0。这种方法在 IGCSE 考试中经常出现，也是快速验证因式分解结果的途径。

7. The Graph of a Quadratic Function | 二次函数的图像

The graph of a quadratic function y = ax² + bx + c is a smooth U-shaped curve called a parabola. The sign of a determines the orientation: when a > 0, the parabola opens upwards, resembling a U; when a < 0, it opens downwards, resembling an inverted U.

二次函数 y = ax² + bx + c 的图像是一条平滑的 U 形曲线，称为抛物线。a 的正负决定开口方向：当 a > 0 时，抛物线开口向上，形如 U；当 a < 0 时，开口向下，形如倒 U。

The graph of a quadratic function has several key features: the vertex (turning point), the axis of symmetry, the y-intercept, and the x-intercepts (if real roots exist). These features allow us to sketch the parabola accurately without plotting many points.

二次函数图像有几个关键特征：顶点（转向点）、对称轴、y 轴截距以及 x 轴截距（若存在实数根）。这些特征使我们无需绘制大量点就能准确画出抛物线草图。

For y = 2x² – 8x + 6, since a = 2 > 0, the parabola opens upward. The y-intercept is the constant term, 6. The x-intercepts are found by solving 2x² – 8x + 6 = 0, which factors to 2(x – 1)(x – 3) = 0, giving x = 1 and x = 3.

对于 y = 2x² – 8x + 6，由于 a = 2 > 0，抛物线开口向上。y 轴截距是常数项 6。x 轴截距通过解 2x² – 8x + 6 = 0 得出，分解为 2(x – 1)(x – 3) = 0，即 x = 1 和 x = 3。

8. The Vertex and Axis of Symmetry | 顶点与对称轴

The vertex is the turning point of the parabola, and the axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two mirror images. For y = ax² + bx + c, the x-coordinate of the vertex is given by x = -b/2a.

顶点是抛物线的转向点，而对称轴是经过顶点的竖直直线，将抛物线分成左右对称的两部分。对于 y = ax² + bx + c，顶点的 x 坐标为 x = -b/2a。

x_vertex = -b / 2a

Once we know the x-coordinate of the vertex, we substitute it back into the original equation to find the y-coordinate. This gives the complete vertex coordinates, which are essential for sketching the graph and solving optimisation problems.

已知顶点的 x 坐标后，将其代回原方程即可求得 y 坐标。这样我们就得到了完整的顶点坐标，这对于绘制图像和解决最优化问题至关重要。

For y = x² – 6x + 5, we have a = 1 and b = -6, so x_vertex = 6/2 = 3. Substituting x = 3 gives y = 9 – 18 + 5 = -4. The vertex is at (3, -4), and the axis of symmetry is x = 3. The minimum value of the function is -4.

对于 y = x² – 6x + 5，a = 1，b = -6，因此 x_vertex = 6/2 = 3。代入 x = 3 得 y = 9 – 18 + 5 = -4。顶点为 (3, -4)，对称轴为 x = 3，函数的最小值为 -4。

9. Solving Quadratic Inequalities | 求解二次不等式

Quadratic inequalities extend the concept of quadratic equations. Instead of finding where ax² + bx + c = 0, we are interested in where the expression is positive or negative relative to zero. This requires understanding the graph of the quadratic function.

二次不等式是二次方程的延伸。我们关心的不是 ax² + bx + c = 0 的解，而是表达式在何处大于零或小于零。这需要理解二次函数的图像。

To solve x² – 3x – 10 < 0, we first factor: (x - 5)(x + 2) < 0. The roots are x = 5 and x = -2. Since the parabola opens upward (a = 1 > 0), the expression is negative between the roots. Therefore, the solution is -2 < x < 5.

解不等式 x² – 3x – 10 < 0，首先因式分解：(x - 5)(x + 2) < 0。根为 x = 5 和 x = -2。由于抛物线开口向上（a = 1 > 0），表达式在两根之间为负。因此，解集为 -2 < x < 5。

For a > 0: ax² + bx + c < 0 when x is between the roots

The sign of the quadratic depends on the interval relative to the roots. A sketch of the parabola is the most reliable way to determine the correct solution interval, especially for more complex inequalities.

二次表达式的符号取决于所在的区间（相对于根的位置）。绘制抛物线草图是确定正确解区间最可靠的方法，尤其是对于更复杂的不等式。

10. Applications in Problem Solving | 实际应用问题

Quadratic equations arise naturally in many real-world contexts, from physics to geometry to economics. IGCSE examinations frequently include word problems that require translating a situation into a quadratic equation, solving it, and interpreting the answers in context.

二次方程在从物理、几何到经济学等许多实际场景中自然出现。IGCSE 考试经常包含应用题，需要将实际情境转化为二次方程，求解并解释答案的实际意义。

Classic examples include: finding the dimensions of a rectangle given its area and perimeter; calculating the time taken for a projectile to reach a certain height; or determining the break-even points in a simple business model. In each case, negative or unrealistic roots must be rejected.

经典例子包括：已知矩形的面积和周长求其尺寸；计算抛射物达到某一高度所需的时间；或确定简单商业模型中的盈亏平衡点。在每种情况下，都需要舍弃负数或不切实际的根。

For example, the height h metres of a ball after t seconds is given by h = 20t – 5t². To find when the ball hits the ground, we set h = 0: 20t – 5t² = 0, which factors to 5t(4 – t) = 0. Thus t = 0 (launch time) or t = 4 seconds. The ball lands after 4 seconds.

例如，球在 t 秒后的高度 h 米由 h = 20t – 5t² 给出。要求球何时落地，令 h = 0：20t – 5t² = 0，分解为 5t(4 – t) = 0。因此 t = 0（出发点）或 t = 4 秒。球在 4 秒后落地。

11. Common Mistakes and How to Avoid Them | 常见错误与避坑指南

Even strong students make predictable errors when working with quadratic equations. Recognising these pitfalls is the first step toward avoiding them in the exam.

即使是优秀的学生在处理二次方程时也会犯一些典型的错误。识别这些陷阱是考试中避免它们的第一步。

Forgetting to set the equation to zero before factorising. Always rearrange into ax² + bx + c = 0 first.

因式分解前忘记将方程移项为零。务必先将方程整理为 ax² + bx + c = 0 的形式。
Dropping a root when taking square roots. Remember that x² = k implies x = ±√k, not just the positive root.

开平方时丢根。记住 x² = k 意味着 x = ±√k，不能只取正根。
Misapplying the quadratic formula by using the wrong signs for b. Substitute carefully and double-check arithmetic.

套用求根公式时 b 的符号出错。代入时要仔细，并复核计算过程。
Forgetting that a = 0 makes the equation linear, not quadratic.

忽略 a = 0 时方程变为一次而非二次。

Practising a variety of problems and reviewing mistakes systematically will build both speed and accuracy. Always verify your solutions by substituting them back into the original equation.

通过练习不同类型的题目并有条理地复盘错误，可以同时提升速度和准确性。始终将解代回原方程进行验证。

12. Summary and Revision Checklist | 总结与复习清单

Quadratic equations are a cornerstone of IGCSE Mathematics. Mastery comes from understanding the standard form, practising all three solution methods (factorisation, completing the square, and the quadratic formula), and being able to interpret roots graphically and algebraically.

二次方程是 IGCSE 数学的基石。熟练掌握来源于：理解标准形式、练习三种解法（因式分解法、配方法、求根公式），以及能够从代数和图像两个角度解释根的意义。

Before the exam, make sure you can:

考试前，请确保你能够：

Identify a quadratic equation and express it in standard form.

识别二次方程并将其写成标准形式。
Solve quadratics by factorisation, completing the square, and the quadratic formula.

使用因式分解法、配方法和求根公式解二次方程。
Use the discriminant to determine the nature of the roots.

使用判别式确定根的性质。
Find the vertex, axis of symmetry, intercepts, and sketch the graph.

求顶点、对称轴、截距并画出图像草图。
Apply the sum and product of roots to form new equations.

利用根的和与积构造新方程。
Solve quadratic inequalities using graphical reasoning.

利用图像推理求解二次不等式。

Practice is the key. Solve at least 10 mixed problems daily for a week before the exam.

练习是关键。考试前一周每天至少完成 10 道混合题型。

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Mastering Quadratic Equations | 掌握二次方程