📚 Solving Quadratic Equations | 解二次方程
A quadratic equation is one of the most frequently tested topics in IGCSE Mathematics. It appears in both Paper 2 and Paper 4, across Core and Extended syllabi. Mastering this topic requires a clear understanding of its forms, solution methods, and the meaning of the discriminant.
二次方程是 IGCSE 数学中最常考的考点之一，在 Paper 2 和 Paper 4 中都会出现，涵盖 Core 和 Extended 两个大纲。掌握这一专题需要清晰地理解其形式、求解方法以及判别式的含义。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation is any equation that can be written in the standard form:
ax² + bx + c = 0, where a ≠ 0
nt of x², b is the coefficient of x, and c is a constant. The condition a ≠ 0 is essential — near. For example, 2x² − 5x + 3 = 0 is a quadratic equation with a = 2, b = −5, and c = 3.
这里，a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 至关重要——如果 a = 0，方程就变成了一次方程。例如，2x² − 5x + 3 = 0 是一个二次方程，其中 a = 2，b = −5，c = 3。
rranged form, such as x² = 4x + 5. Before solving, always rearrange it into standard form:
有时方程以展开或整理后的形式给出，例如 x² = 4x + 5。在求解之前，务必先将其化为标准形式：
x² − 4x − 5 = 0
2. Solving by Factorisation | 因式分解法
best when a = 1. The goal is to express the quadratic as a product of two linear brackets.
因式分解法是 IGCSE 中首先教授的方法，在 a = 1 时最为适用。其目标是将二次式表示为两个一次括号的乘积。
look for two numbers that multiply to give −5 (the constant c) and add to give −4 (the coefficient b). The numbers are −5 and 1:
考虑方程 x² − 4x − 5 = 0。我们需要找到两个数，它们相乘等于 −5（常数项 c），相加等于 −4（一次项系数 b）。这两个数是 −5 和 1：
(x − 5)(x + 1) = 0
For the product of two factors to equal zero, at least one factor must be zero. Therefore:
要使两个因式的乘积为零，至少有一个因式必须为零。因此：
x − 5 = 0 → x = 5 or x + 1 = 0 → x = −1
In an exam, always check that the pair of numbers multiplies to c and adds to b. A common trap is choosing the correct product but the wrong sign combination.
在考试中，务必检查这对数相乘等于 c、相加等于 b。一个常见陷阱是选对了乘积，但符号组合选错。
3. Solving When a ≠ 1 | 当 a ≠ 1 时的因式分解
0. Multiply a and c: 2 × 3 = 6. Now find two numbers whose product is 6 and whose sum is 7 — these are 6 and 1.
当 a ≠ 1 时，因式分解需要更加小心。以方程 2x² + 7x + 3 = 0 为例。将 a 和 c 相乘：2 × 3 = 6。然后找到两个数，其乘积为 6，其和为 7——这两个数是 6 和 1。
Rewrite the middle term using these two numbers:
用这两个数重写中间项：
2x² + 6x + x + 3 = 0
Now factor by grouping the first two terms and the last two terms:
现在对前两项和后两项分别提取公因式：
2x(x + 3) + 1(x + 3) = 0
(x + 3)(2x + 1) = 0
Thus x = −3 or x = −½. This method, often called the "cross method," is reliable for all quadratics with rational roots.
因此 x = −3 或 x = −½。这种方法通常称为"十字相乘法"，对所有具有有理根的二次方程都可靠。
4. Completing the Square | 配方法
+ r = 0. This method is particularly useful when the quadratic does not factorise neatly.
配方法是一种将二次式转化为 p(x + q)² + r = 0 形式的技术。当二次式无法整齐地因式分解时，这种方法尤其有用。
Take x² + 6x + 2 = 0. Halve 6 to get 3, then write:
以 x² + 6x + 2 = 0 为例。将 6 除以 2 得到 3，然后写下：
(x + 3)² − 9 + 2 = 0
(x + 3)² − 7 = 0
Then solve by isolating the squared term:
然后通过分离平方项来求解：
(x + 3)² = 7
x + 3 = ±√7
x = −3 ± √7
g the square root of both sides, the ± sign must be included. The exact answer is x = −3 + √7 or x = −3 − √7.
记住：对两边同时开平方时，必须包含 ± 号。精确答案是 x = −3 + √7 或 x = −3 − √7。
5. The Quadratic Formula | 求根公式
The quadratic formula is the universal method for solving any quadratic equation. It works for all quadratics, including those with irrational or complex roots.
求根公式是解任何二次方程的通用方法。它适用于所有二次方程，包括具有无理根或复数根的方程。
x = (−b ± √b² − 4ac) / 2a
Apply this to 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute into the formula:
将此公式应用于 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入公式：
x = (4 ± √((−4)² − 4 × 2 × (−3))) / (2 × 2)
x = (4 ± √16 + 24) / 4
x = (4 ± √40) / 4 = (4 ± 2√10) / 4 = 1 ± √10/2
In IGCSE exams, the formula is usually provided on the formula sheet, but you must know which values to substitute and how to simplify surds correctly.
在 IGCSE 考试中，公式通常会在公式表中给出，但你必须知道代入哪些值以及如何正确化简根式。
6. The Discriminant | 判别式
The expression b² − 4ac inside the quadratic formula is called the discriminant. It determines the nature of the roots without solving the entire equation.
求根公式中的 b² − 4ac 被称为判别式。它无需解整个方程即可确定根的性质。
If b² − 4ac > 0: the equation has two distinct real roots.
如果 b² − 4ac > 0：方程有两个不同的实数根。
If b² − 4ac = 0: the equation has one repeated real root (a "double root").
如果 b² − 4ac = 0：方程有一个重根（"二重根"）。
If b² − 4ac < 0: the equation has no real roots.
如果 b² − 4ac < 0：方程没有实数根。
is exactly one root. Factorising gives (x − 2)² = 0, confirming x = 2 is a repeated root.
例如，对于 x² − 4x + 4 = 0，判别式为 (−4)² − 4 × 1 × 4 = 16 − 16 = 0，因此恰好有一个根。因式分解得到 (x − 2)² = 0，确认 x = 2 是重根。
7. Solving Word Problems | 应用题求解
Quadratic equations frequently appear in real-world problems, especially geometry and number problems. A typical example: "The area of a rectangle is 24 cm², and its length is 5 cm longer than its width. Find the width."
二次方程经常出现在实际应用题中，尤其是几何和数字问题。一个典型例子："矩形的面积为 24 cm²，其长比宽长 5 cm。求宽。"
Let the width be x. Then the length is x + 5. The area equation is:
设宽为 x，则长为 x + 5。面积方程为：
x(x + 5) = 24
x² + 5x − 24 = 0
(x + 8)(x − 3) = 0
is x = 3 cm. Always check whether both solutions make sense in the context of the problem.
因此 x = −8 或 x = 3。由于宽度不能为负数，唯一有效的答案是 x = 3 cm。务必检查两个解在问题情境中是否都有意义。
8. Graphs of Quadratic Functions | 二次函数图像
nction y = ax² + bx + c always produces a parabola. If a > 0, the parabola opens upward (a "U" shape) and has a minimum point. If a < 0, it opens downward (an "∩" shape) and has a maximum point.
二次函数 y = ax² + bx + c 的图像始终是抛物线。如果 a > 0，抛物线开口向上（"U"形），具有最小值点。如果 a < 0，抛物线开口向下（"∩"形），具有最大值点。
to the x-intercepts of the graph. The x-coordinate of the vertex is given by x = −b/(2a).
方程 ax² + bx + c = 0 的根对应图像与 x 轴的交点。顶点的 x 坐标为 x = −b/(2a)。
If the discriminant is negative, the parabola does not intersect the x-axis at all. If it is zero, the parabola touches the x-axis at exactly one point — the vertex lies on the axis.
如果判别式为负，抛物线完全不与 x 轴相交。如果判别式为零，抛物线恰好与 x 轴相切于一点——顶点在该轴上。
9. Solving Quadratic Inequalities | 解二次不等式
< 0. First solve the corresponding equation to find the critical values x = −1 and x = 5.
一个相关技能是解不等式，如 x² − 4x − 5 < 0。首先解对应的方程，找到临界值 x = −1 和 x = 5。
Since a = 1 > 0, the parabola opens upward and is below the x-axis between the two roots:
画出抛物线 y = x² − 4x − 5 有助于确定哪些区间满足不等式。由于 a = 1 > 0，抛物线开口向上，在两个根之间位于 x 轴下方：
−1 < x < 5
For the inequality x² − 4x − 5 > 0, the solution would be x < −1 or x > 5. In IGCSE, sketching a quick graph is often the fastest and safest way to solve a quadratic inequality.
对于不等式 x² − 4x − 5 > 0，解为 x < −1 或 x > 5。在 IGCSE 中，快速画图常常是解二次不等式最快、最安全的方法。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
plying the formula. Another common mistake is missing the ± sign when taking square roots.
IGCSE 考试中最常见的错误之一是忘记在因式分解或套用公式之前将方程化为标准形式。另一个常见错误是在开平方时漏掉 ± 号。
Always check whether a = 0 — if so, it is not a quadratic equation.
始终检查 a 是否等于 0——如果是，则不是二次方程。
When using the quadratic formula, be careful with negative signs: substitute b as a whole value, including its sign.
使用求根公式时，注意负号：代入 b 时应包含其符号整体代入。
If a question asks for answers to 2 decimal places or 3 significant figures, round only at the final step.
如果题目要求答案保留 2 位小数或 3 位有效数字，只在最后一步进行四舍五入。
In word problems, reject any solution that is not physically possible (e.g., negative lengths).
在应用题中，舍弃任何在物理上不可行的解（例如负长度的解）。
Approximately 30% of students lose marks on quadratic questions because they do not show their working. Always display each step clearly.
大约 30% 的学生因为在二次方程题目上不展示计算过程而丢分。务必清晰地写出每一步。
With consistent practice, recognising the most efficient method for each equation — factorisation for neat rational roots, completing the square for vertex form, and the formula as a universal fallback — will become second nature.
通过持续练习，识别每种方程最高效的方法——有理根较整齐时用因式分解，需要顶点形式时用配方法，而求根公式作为通用备选——将会变得得心应手。
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