📚 IGCSE Maths Teacher Guide G-5-500: Quadratic Equations | IGCSE 数学教师用书 G-5-500：二次方程
Quadratic equations are a central topic in the IGCSE Mathematics syllabus. They appear in algebra, graphs, inequalities and real-world modelling questions. This guide covers the standard form, four solving methods, the discriminant, graphing, turning points and common exam pitfalls. By mastering these ideas, you can tackle a wide range of IGCSE questions with confidence.
二次方程是 IGCSE 数学大纲中的核心主题。它们出现在代数、图像、不等式和实际建模题中。本指南涵盖标准形式、四种求解方法、判别式、图像、顶点以及常见考试陷阱。掌握这些内容后，你就能自信地应对各类 IGCSE 题目。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree two. Its standard form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程是一个二次多项式方程。其标准形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。
ax² + bx + c = 0, a ≠ 0
The expression ax² + bx + c is called a quadratic expression. If a = 0, the equation becomes linear, not quadratic. Quadratic means the highest power of the variable x is 2.
表达式 ax² + bx + c 称为二次式。如果 a = 0，方程就变成线性方程，不再是二次方程。二次意味着变量 x 的最高次数为 2。
In IGCSE exams, you may be asked to write an equation in standard form, identify a, b and c, and then decide which solving method is suitable.
在 IGCSE 考试中，你可能会被要求将方程写成标准形式，指出 a、b、c，并选择适合的求解方法。
2. Solving by Factorising | 因式分解法求解
Factorising is often the quickest method when the quadratic expression can be written as a product of two linear factors. For example, x² + 5x + 6 = (x + 2)(x + 3).
当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。例如，x² + 5x + 6 = (x + 2)(x + 3)。
To solve x² + 5x + 6 = 0, set each factor equal to zero. From x + 2 = 0 we get x = −2, and from x + 3 = 0 we get x = −3. Therefore the solutions are x = −2 or x = −3.
解 x² + 5x + 6 = 0 时，令每个因式等于零。由 x + 2 = 0 得 x = −2，由 x + 3 = 0 得 x = −3。因此解为 x = −2 或 x = −3。
Always make sure the right-hand side is 0 before factorising. If the equation is not equal to zero, rearrange it first.
在因式分解前始终确保等号右边为 0。如果方程不等于零，先移项整理。
- Check for common factors first: 2x² + 4x = 2x(x + 2).
- 先检查公因式：2x² + 4x = 2x(x + 2)。
- Do not divide both sides by x, as this may lose the solution x = 0.
- 不要两边同时除以 x，否则可能丢失 x = 0 这个解。
3. The Quadratic Formula | 求根公式
When factorising is difficult or impossible, use the quadratic formula. It solves any quadratic equation of the form ax² + bx + c = 0.
当因式分解困难或无法进行时，使用求根公式。它可以求解任何形如 ax² + bx + c = 0 的二次方程。
x = (−b ± √(b² − 4ac)) / (2a)
The symbol ± means you must calculate two values: one using the plus sign and one using the minus sign. The expression under the square root, b² − 4ac, is called the discriminant.
符号 ± 表示你必须计算两个值：一个使用加号，一个使用减号。根号下的表达式 b² − 4ac 称为判别式。
For example, solve 2x² − 4x − 6 = 0. Here a = 2, b = −4, c = −6. Substituting gives x = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4, so x = 3 or x = −1.
例如，解 2x² − 4x − 6 = 0。这里 a = 2，b = −4，c = −6。代入得 x = (4 ± √(16 + 48)) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4，所以 x = 3 或 x = −1。
4. Completing the Square | 配方法
Completing the square rewrites a quadratic as a squared binomial plus a constant. This is useful for solving, graphing and finding the vertex.
配方法将二次式改写为一个二项式的平方加上一个常数。这对求解、绘图以及求顶点都很有用。
Start with x² + 6x + 5. Take half the coefficient of x, which is 3, and square it to get 9. Then write x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.
以 x² + 6x + 5 为例。取 x 系数的一半，即 3，再平方得 9。然后写成 x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。
To solve x² + 6x + 5 = 0, set (x + 3)² − 4 = 0, so (x + 3)² = 4. Taking square roots gives x + 3 = ±2, hence x = −1 or x = −5.
解 x² + 6x + 5 = 0 时，令 (x + 3)² − 4 = 0，因此 (x + 3)² = 4。开平方得 x + 3 = ±2，所以 x = −1 或 x = −5。
When the coefficient of x² is not 1, factor it out first before completing the square.
当 x² 的系数不是 1 时，先提出该系数，再进行配方。
5. The Discriminant | 判别式
The discriminant is given by Δ = b² − 4ac. It tells us the nature of the roots without actually solving the equation.
判别式由 Δ = b² − 4ac 给出。它无需实际求解方程即可说明根的性质。
| Discriminant | Nature of roots | 根的性质 |
| Δ > 0 | two distinct real roots | 两个不相等的实根 |
| Δ = 0 | one repeated real root | 一个重根（两个相等的实根） |
| Δ < 0 | no real roots | 没有实根 |
For example, for x² − 4x + 13 = 0, the discriminant is (−4)² − 4(1)(13) = 16 − 52 = −36. Since Δ < 0, the equation has no real roots.
例如，对于 x² − 4x + 13 = 0，判别式为 (−4)² − 4(1)(13) = 16 − 52 = −36。因为 Δ < 0，该方程没有实根。
6. Graphs of Quadratic Functions | 二次函数图像
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards. If a < 0, it opens downwards.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上；如果 a < 0，开口向下。
The y-intercept is the constant term c, because when x = 0, y = c. The x-intercepts are the real roots of ax² + bx + c = 0.
y 轴截距是常数项 c，因为当 x = 0 时，y = c。x 轴截距是 ax² + bx + c = 0 的实根。
The axis of symmetry is a vertical line that passes through the vertex. Its equation is x = −b / (2a).
对称轴是一条经过顶点的竖直直线。其方程为 x = −b / (2a)。
Axis of symmetry: x = −b / (2a)
Sketching a parabola in IGCSE often requires you to label the y-intercept, any x-intercepts and the vertex.
在 IGCSE 中画抛物线草图时，通常需要标出 y 轴截距、任意 x 轴截距以及顶点。
7. Turning Points and Symmetry | 顶点与对称性
The vertex of a parabola is its turning point. For a > 0, it is a minimum point. For a < 0, it is a maximum point.
抛物线的顶点是其转折点。当 a > 0 时，它是最小值点；当 a < 0 时，它是最大值点。
In completed square form y = a(x − h)² + k, the vertex is simply (h, k). This is why completing the square is so useful for graphing.
在配方法形式 y = a(x − h)² + k 中，顶点就是 (h, k)。这就是配方法在绘图时如此有用的原因。
If the equation is given in standard form, find the x-coordinate of the vertex with x = −b / (2a), then substitute this value back into the equation to find the y-coordinate.
如果给出的是标准形式，先用 x = −b / (2a) 求出顶点的 x 坐标，再将该值代回方程求出 y 坐标。
The parabola is symmetric about its axis of symmetry. This symmetry can be used to check your sketch or find missing points.
抛物线关于其对称轴对称。这种对称性可以用来检查你的草图或找出缺失的点。
8. Quadratic Inequalities | 二次不等式
To solve a quadratic inequality such as ax² + bx + c > 0 or ax² + bx + c < 0, first find the roots of the related quadratic equation.
要解形如 ax² + bx + c > 0 或 ax² + bx + c < 0 的二次不等式，首先求出相应二次方程的根。
For example, solve x² − 4x + 3 > 0. The roots are x = 1 and x = 3. These divide the number line into three intervals: x < 1, 1 < x < 3 and x > 3.
例如，解 x² − 4x + 3 > 0。根为 x = 1 和 x = 3。它们将数轴分成三个区间：x < 1，1 < x < 3 和 x > 3。
Test a value from each interval in the original inequality. The parabola is above the x-axis when x < 1 or x > 3, so the solution is x < 1 or x > 3.
在每个区间中取一个值代入原不等式检验。当 x < 1 或 x > 3 时，抛物线位于 x 轴上方，因此解为 x < 1 或 x > 3。
x ∈ (−∞, 1) ∪ (3, ∞)
Always pay attention to whether the inequality is strict (> or <) or inclusive (≥ or ≤). Strict inequalities exclude the roots; inclusive inequalities include the roots.
始终注意不等式是严格（> 或 <）还是包含（≥ 或 ≤）。严格不等式不包括根；包含不等式包括根。
9. Real-World Applications | 实际应用
Quadratic equations model many real-life situations, including projectile motion, area optimisation and profit analysis. For example, the height of a ball thrown upwards might be modelled by h(t) = −5t² + 20t + 1, where h is height in metres and t is time in seconds.
二次方程可以模拟许多现实情境，包括抛体运动、面积优化和利润分析。例如，向上抛出的球的高度可建模为 h(t) = −5t² + 20t + 1，其中 h 为高度（米），t 为时间（秒）。
To find when the ball hits the ground, set h(t) = 0 and solve the quadratic equation. In context, negative time solutions are usually rejected because time cannot be negative.
要求球何时落地，令 h(t) = 0 并求解二次方程。在具体情境中，负时间解通常舍去，因为时间不能为负。
Area problems often lead to quadratic equations. For instance, if a rectangular field has length x + 3 and width x, and its area is 40, then x(x + 3) = 40, which expands to x² + 3x − 40 = 0.
面积问题常常引出二次方程。例如，一个矩形场地的长为 x + 3，宽为 x，面积为 40，则 x(x + 3) = 40，展开得 x² + 3x − 40 = 0。
Always interpret your mathematical solution in the original context. Check that lengths, times and quantities are positive and realistic.
始终在原始情境中解释你的数学解。检查长度、时间和数量是否为正且符合实际。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
One of the most common IGCSE mistakes is forgetting to set the equation equal to zero before factorising. Always rearrange first.
IGCSE 中最常见的错误之一是因式分解前忘记让方程等于零。务必先移项整理。
Another frequent error is dividing both sides of a quadratic equation by x. This can remove the solution x = 0. Instead, factor the x out and use the zero product property.
另一个常见错误是两边同时除以 x。这可能会去掉 x = 0 这个解。正确做法是将 x 提出来，并利用零积性质。
Do not forget the ± sign in the quadratic formula. Missing it gives only one root instead of two. Also, be careful with negative values of b: subtracting a negative changes the sign.
不要忘记求根公式中的 ± 号。漏掉它只会得到一个根，而不是两个。同时，注意 b 为负值的情况：减去一个负数会改变符号。
In IGCSE exams, marks are awarded for method as well as final answers. Always show your substitution, simplification and checking steps.
在 IGCSE 考试中，方法分和最终答案分都会给。始终展示你的代入、化简和检验步骤。
Finally, practise past paper questions under timed conditions. Become familiar with the syllabus formula sheet and know when to use each method: factorising, formula, completing the square or discriminant.
最后，在限时
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