📚 Mastering Quadratic Equations for IGCSE Mathematics | 攻克 IGCSE 数学中的二次方程
Quadratic equations are a central topic in IGCSE Mathematics. They appear in Paper 2 and Paper 4, either as direct solving questions or as part of graph work, word problems, and algebra-related questions. Understanding the different solving techniques and the properties of quadratic functions will give you a strong advantage.
二次方程是 IGCSE 数学的核心考点。无论是在 Paper 2 还是 Paper 4 中，它们都会以直接求解题、图像题、应用题或代数综合题的形式出现。掌握不同的解法以及二次函数的性质，能让你在考试中占据明显优势。
1. Standard Form of a Quadratic Equation | 二次方程的标准形式
A quadratic equation in one variable can be written in the standard form ax2 + bx + c = 0 , where a, b, c are real numbers and a ≠ 0 . The condition a ≠ 0 is essential because if a = 0 , the equation becomes linear, not quadratic.
一元二次方程可以写成标准形式 ax2 + bx + c = 0 ，其中 a, b, c 是实数，且 a ≠ 0 。 a ≠ 0 这一条件至关重要，因为如果 a = 0 ，方程就变成了一次方程，而不是二次方程。
For example, 2x2 - 3x + 1 = 0 is a quadratic equation, while 2x - 3 = 0 is not. In IGCSE questions, you must recognise quadratic equations even when they are not given in standard form, such as 3x2 = 2x + 5 , which can be rearranged to 3x2 - 2x - 5 = 0 .
例如， 2x2 - 3x + 1 = 0 是二次方程，而 2x - 3 = 0 不是。在 IGCSE 题目中，你需要能够识别未写成标准形式的二次方程，例如 3x2 = 2x + 5 ，它可以变形为 3x2 - 2x - 5 = 0 。
2. The Discriminant | 判别式
The discriminant is a key concept used to determine the nature of the roots without solving the equation. For the quadratic equation ax2 + bx + c = 0 , the discriminant is defined as:
判别式是无需解方程即可判断根的性质的重要概念。对于二次方程 ax2 + bx + c = 0 ，判别式定义为：
Δ = b² − 4ac
The value of Δ tells us how many real roots the equation has:
Δ 的值告诉我们方程有多少个实数根：
| Value of Δ | Nature of Roots |
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
This quick check is especially useful in multiple-choice and short-answer questions.
这种快速判断在选择题和简答题中尤其有用。
3. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when the quadratic equation has simple integer roots. The idea is to write the quadratic as a product of two linear factors.
当二次方程具有简单的整数根时，因式分解往往是最快的方法。其核心思想是将二次式写成两个一次因式的乘积。
For example, solve x2 - 5x + 6 = 0 . We look for two numbers that multiply to 6 and add to −5. These are −2 and −3, so:
例如，解 x2 - 5x + 6 = 0 。我们要找两个数，它们相乘为 6，相加为 −5。这两个数是 −2 和 −3，因此：
(x − 2)(x − 3) = 0
Setting each factor to zero gives x = 2 or x = 3 .
令每个因式等于零，得到 x = 2 或 x = 3 。
If the coefficient of x2 is not 1, you may need to factorise by grouping or use trial and improvement. However, not every quadratic can be factorised using simple integers, so other methods are needed.
如果 x2 的系数不是 1，你可能需要使用分组分解或尝试法。但并非所有二次方程都能用简单整数因式分解，因此还需要掌握其他方法。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)2 + q . This form is especially useful for finding the turning point of a quadratic graph and for solving equations.
配方法将二次式改写为 (x + p)2 + q 的形式。这种形式尤其适用于寻找二次函数图像的顶点，以及求解方程。
Consider x2 + 6x + 1 = 0 . Take half of 6, which is 3, and square it to get 9. Then:
考虑方程 x2 + 6x + 1 = 0 。取 6 的一半为 3，平方得到 9。于是：
(x + 3)² − 9 + 1 = 0
(x + 3)² − 8 = 0
So (x + 3)2 = 8 , giving x + 3 = ±√8 , hence x = −3 ± 2√2 .
因此 (x + 3)2 = 8 ，得到 x + 3 = ±√8 ，所以 x = −3 ± 2√2 。
When the coefficient of x2 is not 1, first factor it out before completing the square.
当 x2 的系数不为 1 时，先将其提取出来，再进行配方。
5. The Quadratic Formula | 二次求根公式
The quadratic formula works for every quadratic equation and is particularly reliable when factorisation is difficult. For ax2 + bx + c = 0 :
二次求根公式适用于所有二次方程，尤其在因式分解困难时非常可靠。对于 ax2 + bx + c = 0 ：
x = (−b ± √b² − 4ac) / (2a)
Take the equation 2x2 - 3x - 5 = 0 . Here a = 2, b = -3, c = -5 . Substitute into the formula:
以方程 2x2 - 3x - 5 = 0 为例。这里 a = 2, b = -3, c = -5 。代入公式：
x = (−(−3) ± √((−3)² − 4 × 2 × (−5))) / (2 × 2)
x = (3 ± √9 + 40) / 4 = (3 ± √49) / 4
Thus x = (3 + 7)/4 = 2.5 or x = (3 - 7)/4 = -1 .
因此 x = (3 + 7)/4 = 2.5 或 x = (3 - 7)/4 = -1 。
Always check whether the question expects exact values or decimal answers. In the non-calculator paper, you may need to leave answers in surd form.
注意题目要求精确值还是近似值。在不使用计算器的试卷中，你可能需要保留根号形式。
6. Using the Discriminant to Solve Problems | 利用判别式解决问题
Beyond classifying roots, the discriminant can be used to find unknown parameters. For instance, if the equation x2 + kx + 4 = 0 has exactly one real root, then the discriminant must be zero.
除了判断根的类型外，判别式还可用于求未知参数。例如，若方程 x2 + kx + 4 = 0 只有一个实数根，那么判别式必须为零。
k² − 4 × 1 × 4 = 0
So k2 = 16 , giving k = ±4 . This type of question tests both the discriminant concept and algebraic manipulation.
因此 k2 = 16 ，得到 k = ±4 。这类题同时考查判别式概念和代数运算能力。
In IGCSE, you may also be asked to prove that a quadratic has no real roots by showing Δ < 0.
在 IGCSE 考试中，还常要求通过证明 Δ < 0 来表明一个二次方程没有实数根。
7. Graphs of Quadratic Functions | 二次函数图像
The graph of a quadratic function y = ax2 + bx + c is a parabola. If a > 0 , it opens upward; if a < 0 , it opens downward. The curve is symmetric about a vertical line called the axis of symmetry.
二次函数 y = ax2 + bx + c 的图像是一条抛物线。若 a > 0 ，抛物线开口向上；若 a < 0 ，则开口向下。曲线关于一条竖直线对称，这条线称为对称轴。
To sketch the graph, you need to find the roots, the y-intercept, the turning point, and the axis of symmetry. The y-intercept is simply c , and the x-intercepts are the roots of the equation ax2 + bx + c = 0 .
画图时需要找出根、y 轴截距、顶点和对称轴。y 轴截距就是 c ，而 x 轴截距是方程 ax2 + bx + c = 0 的根。
Axis of symmetry: x = −b / (2a)
The axis of symmetry is the vertical line through the turning point.
对称轴是经过顶点的那条竖直线。
8. Vertex Form and Transformations | 顶点式与图像变换
Any quadratic function can be written in the vertex form y = a(x - h)2 + k , where the vertex is at (h, k) . This form makes it easy to see horizontal and vertical translations.
任何二次函数都可以写成顶点式 y = a(x - h)2 + k ，其中顶点坐标为 (h, k) 。这种形式可以直观地看出水平和垂直平移。
For example, y = 2(x - 3)2 + 4 has a minimum point at (3, 4). Compared with y = x2 , the graph is scaled vertically by 2, shifted right by 3 units, and shifted up by 4 units.
例如， y = 2(x - 3)2 + 4 的最小值点为 (3, 4)。与 y = x2 相比，图像纵向拉伸为原来的 2 倍，向右平移 3 个单位，再向上平移 4 个单位。
You should also recognise that the value of a determines the width and direction of the parabola. A larger absolute value of a makes the parabola narrower.
你还应知道 a 的值决定了抛物线的开口方向和宽窄。 |a| 越大，抛物线越窄。
9. Solving Word Problems with Quadratics | 二次方程应用题
Quadratics are often used to model real-life situations such as areas, projectile motion, and number problems. The key is to translate the information into a quadratic equation correctly.
二次方程常用于建模实际情境，如面积问题、抛体运动问题和数字问题。关键是将题目信息正确转化为二次方程。
Example: The length of a rectangle is 3 cm longer than its width, and its area is 40 cm². Let the width be x cm. Then the length is x + 3 , and the area gives:
例如：一个长方形的长比宽长 3 cm，面积为 40 cm²。设宽为 x cm，则长为 x + 3 ，由面积可得：
x(x + 3) = 40
x² + 3x − 40 = 0
Factorising: (x + 8)(x - 5) = 0 , so x = -8 or x = 5 . Since a width cannot be negative, x = 5 . Therefore the width is 5 cm and the length is 8 cm.
因式分解： (x + 8)(x - 5) = 0 ，所以 x = -8 或 x = 5 。由于宽度不能为负数，取 x = 5 。因此宽为 5 cm，长为 8 cm。
Always reject any solution that does not make sense in the context of the problem.
务必舍去在实际情境中不合理的解。
10. Common Pitfalls and Exam Tips | 常见错误与考试技巧
Many students make avoidable errors in quadratic questions. Here are some important tips:
很多学生在二次方程题目中会犯一些可以避免的错误。以下是一些重要提示：
Always rearrange the equation into standard form before solving.
求解前一定要将方程化为标准形式。
Do not forget to set the equation equal to zero when using the quadratic formula.
使用求根公式时，不要忘记让方程等于零。
Check your solutions by substituting them back into the original equation.
将解代回原方程进行验算。
When using factorisation, make sure the factors multiply to give the constant term and add to give the coefficient of x .
使用因式分解时，确保因式的乘积等于常数项，且相加等于一次项系数。
In graph questions, use the discriminant to check whether the graph crosses the x-axis before sketching.
在图像题中，画图前先用判别式判断图像是否与 x 轴相交。
By practising these methods regularly, you will build the confidence needed for IGCSE examinations.
通过定期练习这些方法，你将为 IGCSE 考试建立足够的信心。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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