Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程

📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程

Quadratic equations are one of the most important topics in the Cambridge IGCSE Mathematics syllabus. They appear in Paper 2 and Paper 4, both as standalone questions and embedded in problems on areas, motion, finance and geometry.

二次方程是剑桥 IGCSE 数学大纲中最重要的主题之一。它们出现在试卷 2 和试卷 4 中，既以独立题目出现，也嵌入到面积、运动、财务和几何问题中。

Mastering the main solution methods will give you confidence and save time in the exam, because many questions combine algebra with interpretation of roots.

掌握主要解法将让你在考试中更有信心并节省时间，因为许多题目将代数与根的解读结合在一起。

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

m ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of x is 2.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数且 a ≠ 0。x 的最高次数是 2。

The condition a ≠ 0 is essential. If a = 0, the equation becomes linear, not quadratic.

条件 a ≠ 0 至关重要。如果 a = 0，方程就变成了一次方程，而不是二次方程。

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

nto standard form ax² + bx + c = 0. Collect all terms on one side and simplify like terms.

在求解之前，一定要将方程整理成标准形式 ax² + bx + c = 0。把所有项移到一边并合并同类项。

2 = 0 has a = 3, b = −7 and c = 2. Careful handling of negative signs is a key exam skill.

例如，3x² − 7x + 2 = 0 中 a = 3、b = −7、c = 2。仔细处理负号是一项关键的考试技能。

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3. Solving by Factorisation | 因式分解法

actors. Write the expression as (px + q)(rx + s) = 0, then set each bracket equal to zero.

当二次式有简单因式时，因式分解是最快的方法。把表达式写成 (px + q)(rx + s) = 0，然后令每个括号等于零。

Example: x² − 5x + 6 = 0 factorises to (x − 2)(x − 3) = 0, so x = 2 or x = 3.

例子：x² − 5x + 6 = 0 因式分解为 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。

The logic is that if a product is zero, at least one factor must be zero. This is called the zero product property.

其逻辑是：如果乘积为零，至少有一个因式必须为零。这称为零乘积性质。

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4. Solving by Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful when the quadratic does not factorise neatly.

配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式。当二次式不能整齐因式分解时，这种方法很有用。

6x + 5 = 0, half of 6 is 3, so write (x + 3)² − 9 + 5 = 0. This becomes (x + 3)² − 4 = 0.

对于 x² + 6x + 5 = 0，6 的一半是 3，因此写成 (x + 3)² − 9 + 5 = 0。整理得到 (x + 3)² − 4 = 0。

Then (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.

然后 (x + 3)² = 4，所以 x + 3 = ±2，得到 x = −1 或 x = −5。

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5. Solving by the Quadratic Formula | 求根公式法

The quadratic formula works for every quadratic equation, even when factorisation is difficult or impossible.

求根公式适用于每一个二次方程，即使因式分解很困难或不可能。

If ax² + bx + c = 0, then:

若 ax² + bx + c = 0，则：

x = (−b ± √b² − 4ac) ÷ (2a)

Substitute the values of a, b and c carefully, especially when they are negative. Use brackets around negative numbers when entering them into a calculator.

代入 a、b、c 的值时要仔细，尤其是负数。在计算器中输入负数时要用括号。

Example: For 2x² − 4x − 3 = 0, a = 2, b = −4, c = −3 gives x = (4 ± √16 + 24) ÷ 4 = (4 ± √40) ÷ 4.

例子：对于 2x² − 4x − 3 = 0，a = 2，b = −4，c = −3，得到 x = (4 ± √16 + 24) ÷ 4 = (4 ± √40) ÷ 4。

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

The discriminant is the expression b² − 4ac inside the square root of the quadratic formula. It tells you the nature of the roots without solving the equation fully.

判别式是求根公式中平方根内的表达式 b² − 4ac。它可以在不完全解方程的情况下告诉你根的性质。

• If b² − 4ac > 0, there are two distinct real roots.

  如果 b² − 4ac > 0，则有两个不同的实根。

• If b² − 4ac = 0, there is one repeated real root.

  如果 b² − 4ac = 0，则有一个重实根。

• If b² − 4ac < 0, there are no real roots.

  如果 b² − 4ac < 0，则没有实根。

Questions often ask you to find the value of k for which a quadratic has equal roots. Set the discriminant equal to zero and solve.

题目经常要求你求出使二次方程有等根的 k 值。令判别式等于零并求解即可。

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7. Graphical Interpretation of Roots | 根的图像意义

e graph y = ax² + bx + c. These are the points where the curve cuts or touches the x-axis.

ax² + bx + c =

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