📚 Quadratic Equations: Factorising, Formula and Graphs | 二次方程：因式分解、公式与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from basic factorising to graphing parabolas and solving real-world problems. This revision guide explains the key methods, common errors, and exam techniques you need to master the topic with confidence.
二次方程贯穿 IGCSE 数学课程，从基础因式分解到抛物线作图以及解决实际问题。本复习指南讲解关键方法、常见错误和考试技巧，帮助你自信掌握这一主题。
1. What is a Quadratic Equation? | 什么是二次方程？
a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.
二次方程是次数为 2 的多项式方程。它可写成一般形式 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，方程变为一次方程，不再是二次方程。
The highest power of the variable x is 2, which gives the equation its characteristic U-shaped graph called a parabola. Every quadratic equation can have two solutions, one solution, or no real solutions depending on its coefficients.
变量 x 的最高次数为 2，这使其图像呈现特有的 U 形，称为抛物线。根据系数的不同，每个二次方程可能有两个解、一个解或没有实数解。
2. Standard Form and Key Terms | 标准形式与关键术语
the quadratic coefficient, b is the linear coefficient, and c is the constant term. The solutions of the equation are also called roots or x-intercepts because they are the x-values where the parabola crosses the x-axis.
在标准形式 ax² + bx + c = 0 中，a 是二次项系数，b 是一次项系数，c 是常数项。方程的解也称为根或 x 轴截距，因为它们是抛物线与 x 轴相交处的 x 值。
- a: quadratic coefficient | 二次项系数
- b: linear coefficient | 一次项系数
- c: constant term | 常数项
- roots: solutions of the equation | 根：方程的解
A quadratic equation can have two distinct real roots, one repeated real root, or no real roots. Recognising the standard form is essential before choosing a solution method.
二次方程可能有两个不相等的实根、一个重根或没有实根。在选择求解方法之前，识别标准形式至关重要。
3. Solving by Factorising | 因式分解法
ctorised as (x + 2)(x + 3) = 0. Setting each bracket equal to zero gives x = -2 or x = -3.
当二次式可写成两个一次因式的乘积时，适合使用因式分解法。例如，x² + 5x + 6 = 0 可分解为 (x + 2)(x + 3) = 0。令每个括号为零，得到 x = -2 或 x = -3。
x² + 5x + 6 = (x + 2)(x + 3) = 0
This method is quick, but it only works neatly when the roots are rational numbers. If you cannot find integer factors within a reasonable time, switch to the quadratic formula or completing the square.
此方法速度快，但仅在根为有理数时较为整洁。若无法在合理时间内找到整数因式，可改用求根公式或配方法。
4. The Quadratic Formula | 求根公式
s difficult or impossible. For ax² + bx + c = 0, the roots are given by the formula below.
求根公式可以解任何二次方程，即使因式分解很困难或无法进行。对于 ax² + bx + c = 0，根由下式给出。
x = (−b ± √b² − 4ac) / 2a
wo solutions: one using the plus sign and one using the minus sign. The expression b² − 4ac under the square root is called the discriminant. Always write the equation in standard form before substituting values into the formula.
符号 ± 表示通常有两个解：一个取加号，一个取减号。平方根下的表达式 b² − 4ac 称为判别式。在代入公式前，务必先将方程写成标准形式。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x − h)² + k. For example, x² + 6x + 5 can be written as (x + 3)² − 4. This form is useful for finding the turning point of a parabola and for solving equations when factorising is not obvious.
配方法将二次式改写为 a(x − h)² + k 的形式。例如，x² + 6x + 5 可写为 (x + 3)² − 4。这种形式便于求抛物线的顶点，也适用于因式分解不明显时的方程求解。
x² + 6x + 5 = (x + 3)² − 4
e square, set (x + 3)² − 4 = 0, then (x + 3)² = 4, so x + 3 = ±2, giving x = -1 or x = -5.
用配方法解 x² + 6x + 5 = 0 时，令 (x + 3)² − 4 = 0，则 (x + 3)² = 4，因此 x + 3 = ±2，得到 x = -1 或 x = -5。
6. Discriminant and Nature of Roots | 判别式与根的性质
peated real root. If D < 0, there are no real roots, but two complex roots exist at a higher level.
判别式 D = b² − 4ac 能告诉我们二次方程有多少个实根。若 D > 0，则有两个不相等的实根。若 D = 0，则有一个重根。若 D < 0，则没有实根，但在更高层次数学中存在两个复根。
| Condition | 条件 | Number of real roots | 实根数量 |
| D > 0 | 2 distinct real roots | 两个不等实根 |
| D = 0 | 1 repeated real root | 一个重根 |
| D < 0 | 0 real roots | 无实根 |
Using the discriminant before solving can save time, especially in multiple-choice questions where you only need to know the number of solutions.
在求解前使用判别式可以节省时间，尤其是在只需知道解数量的选择题中。
7. Graphs of Quadratic Functions | 二次函数图像
. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上，存在最低点。若 a < 0，开口向下，存在最高点。
x + c = 0. If the discriminant is negative, the parabola does not cross the x-axis at all.
y 轴截距为 (0, c)，x 轴截距是方程 ax² + bx + c = 0 的实根。如果判别式为负，抛物线完全不穿过 x 轴。
8. Turning Point and Axis of Symmetry | 顶点与对称轴
inate is x = −b / 2a. The axis of symmetry is the vertical line through the turning point.
顶点是抛物线改变方向的点，其 x 坐标为 x = −b / 2a。对称轴是经过顶点的竖直直线。
x = −b / 2a
ply (h, k). For example, y = (x + 3)² − 4 has vertex (-3, -4) and axis of symmetry x = -3.
在配方式 a(x − h)² + k 中，顶点为 (h, k)。例如，y = (x + 3)² − 4 的顶点为 (-3, -4)，对称轴为 x = -3。
9. Real-World Applications | 实际应用
Quadratic equations model many real-life situations, such as the height of a ball thrown upward, the area of a rectangle with a fixed perimeter, and profit functions in business. By setting the quadratic expression equal to zero, you can find landing times, break-even points, or maximum dimensions.
二次方程可模拟许多现实情境，如向上抛出的球的高度、固定周长矩形的面积以及商业中的利润函数。将二次式设为零，可求出落地时间、盈亏平衡点或最大尺寸。
For example, a ball' e given by h = -5t² + 20t + 1. Setting h = 0 gives the time when the ball hits the ground.
例如，一个球在 t 秒后的高度 h 米可能满足 h = -5t² + 20t + 1。令 h = 0 可求出球落地时间。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
using the quadratic formula. Always check your solutions by substituting them back into the original equation.
常见错误包括在因式分解前忘记将方程设为零，以及在使用求根公式时遗漏 ± 号。务必通过将解代回原方程来检验。
- Forgetting a ≠ 0 in standard form | 忘记标准形式中 a ≠ 0
- Losing the ± sign in the formula | 遗漏公式中的 ± 号
- Not simplifying the square root | 未化简平方根
- Forgetting to check answers | 忘记检验答案
In exams, show all steps clearly because method marks are awarded even if the final answer is incorrect. Always write the formula first, then substitute values carefully.
考试中要清晰展示所有步骤，因为即使最终答案错误，也会给步骤分。务必先写出公式，再仔细代入数值。
11. Worked Example | 例题精讲
Here a = 2, b = −3, c = −5. The discriminant is b² − 4ac = (−3)² − 4(2)(−5) = 9 + 40 = 49.
用求根公式解方程 2x² − 3x − 5 = 0。此处 a = 2，b = −3，c = −5。判别式为 b² − 4ac = (−3)² − 4(2)(−5) = 9 + 40 = 49。
Since D > 0, there are two real roots. Substituting into the formula gives:
由于 D > 0，方程有两个实根。代入公式得到：
x = (3 ± √49) / 4 = (3 ± 7) / 4
(3 + 7)/4 = 10/4 = 5/2 or x = (3 − 7)/4 = −4/4 = −1. The two roots are x = 5/2 and x = −1.
因此 x = (3 + 7)/4 = 10/4 = 5/2 或 x = (3 − 7)/4 = −4/4 = −1。两个根为 x = 5/2 和 x = −1。
12. Summary | 总结
be solved by factorising, completing the square, or using the quadratic formula. The discriminant determines the nature of the roots. Quadratic functions produce parabolas whose key features include the axis of symmetry, vertex, and intercepts.
二次方程 ax² + bx + c = 0 可通过因式分解、配方法或求根公式求解。判别式决定根的性质。二次函数产生抛物线，其关键特征包括对称轴、顶点和截距。
Mastering these methods prepares you well for IGCSE exam questions, especially those involving graphs and real-world contexts. Practise each method and check your answers to build speed and accuracy.
掌握这些方法能帮助你更好地应对 IGCSE 考试题目，尤其是涉及图像和现实情境的题目。练习每种方法并检查答案，以提高速度和准确率。
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