📚 Mastering Quadratic Equations | 掌握二次方程
Quadratic equations and their graphs are a cornerstone of the IGCSE Mathematics syllabus. Whether you are solving an equation, sketching a parabola, or interpreting a real-world model, a deep understanding of quadratics is essential for success in both Paper 2 and Paper 4.
二次方程及其图像是 IGCSE 数学课程的核心内容。无论是解方程、画抛物线草图，还是解释现实世界中的数学模型，深入理解二次方程都是你在 Paper 2 和 Paper 4 中取得成功的必要条件。
1. General Form | 一般形式
A quadratic expression or equation is defined by the highest power of the variable being 2. The standard general form is written as:
二次表达式或方程的定义是变量的最高次数为 2。其标准一般形式可写为：
ax² + bx + c = 0, where a ≠ 0
Here, a, b, and c are constants, and x is the variable. The value of a must never be zero; if a = 0, the equation becomes linear, not quadratic.
这里的 a、b、c 均为常数，x 为变量。其中 a 的值绝不能为零；如果 a = 0，方程将变为线性方程，而非二次方程。
For example, 2x² + 3x - 5 = 0 is a quadratic equation in general form. In this case, a = 2, b = 3, and c = -5.
例如，2x² + 3x - 5 = 0 就是一个一般形式的二次方程。这里 a = 2，b = 3，c = -5。
2. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method when the quadratic has simple integer roots. The goal is to express the equation as a product of two binomials.
当二次方程具有简单的整数根时，因式分解法通常是最快捷的方法。其目标是将方程表示为两个二项式的乘积。
If ax² + bx + c = (px + q)(rx + s) = 0, then x = -q/p or x = -s/r
For example, solve x² - 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. Thus:
例如，解 x² - 5x + 6 = 0。我们寻找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3。因此：
(x - 2)(x - 3) = 0 ⇒ x = 2 or x = 3
Always expand your factorised form to verify your answer. Factorisation becomes more challenging when a ≠ 1, but the underlying principle remains the same: find two numbers whose product is ac and whose sum is b.
务必展开因式分解后的形式来验证答案。当 a ≠ 1 时，因式分解会更具挑战性，但其基本原理相同：寻找两个数，使它们的乘积为 ac，和为 b。
3. Solving by Completing the Square | 配方法
Completing the square transforms a quadratic into a perfect square plus a constant. This method is especially useful when the quadratic cannot be factorised.
配方法将二次方程转化为一个完全平方加上一个常数。当二次方程无法因式分解时，此方法尤其有用。
x² + bx = (x + b/2)² - (b/2)²
For example, solve x² + 6x + 7 = 0. First, complete the square for x² + 6x:
例如，解 x² + 6x + 7 = 0。首先对 x² + 6x 配方：
(x + 3)² - 9 + 7 = 0 ⇒ (x + 3)² - 2 = 0
Then rearrange: (x + 3)² = 2. Taking square roots gives x + 3 = ±√2, so x = -3 ± √2.
然后重新整理：(x + 3)² = 2。两边开平方得 x + 3 = ±√2，所以 x = -3 ± √2。
This method also directly reveals the turning point of the graph, which we will discuss later.
这种方法还能直接揭示图像的顶点坐标，我们将在后面讨论。
4. Solving by the Quadratic Formula | 公式法
The quadratic formula is a universal method that works for every quadratic equation, regardless of whether it factorises. It is particularly useful when the roots are irrational or complex.
二次公式是适用于所有二次方程的通用方法，无论其是否可以因式分解。当根为无理数或复数时，它尤其有用。
x = (-b ± √b² - 4ac) / (2a)
For example, solve 3x² - 4x - 2 = 0. Here a = 3, b = -4, c = -2. Substitute into the formula:
例如，解 3x² - 4x - 2 = 0。这里 a = 3，b = -4，c = -2。代入公式：
x = (4 ± √16 + 24) / 6 = (4 ± √40) / 6 = (4 ± 2√10) / 6
Simplify to x = (2 ± √10) / 3. Always check whether the calculator is allowed; in a non-calculator paper, leave your answer in exact surd form.
化简得 x = (2 ± √10) / 3。务必注意是否允许使用计算器；在不允许使用计算器的试卷中，请将答案保留为精确的根式形式。
5. The Discriminant | 判别式
The discriminant is the part of the quadratic formula under the square root: b² - 4ac. It tells us the nature of the roots without fully solving the equation.
判别式是二次公式中根号内的部分：b² - 4ac。它无需完全解方程就能告诉我们根的性质。
- If b² - 4ac > 0, there are two distinct real roots.
- If b² - 4ac = 0, there is exactly one real root (a repeated root).
- If b² - 4ac < 0, there are no real roots (two complex roots).
- 若 b² - 4ac > 0，则有两个不同的实数根。
- 若 b² - 4ac = 0，则有一个实数根（重根）。
- 若 b² - 4ac < 0，则没有实数根（有两个复数根）。
For example, the equation 2x² - 4x + 3 = 0 has discriminant (-4)² - 4×2×3 = 16 - 24 = -8 < 0, so it has no real roots.
例如，方程 2x² - 4x + 3 = 0 的判别式为 (-4)² - 4×2×3 = 16 - 24 = -8 < 0，因此它没有实数根。
6. Quadratic Graphs | 二次函数图像
The graph of a quadratic function y = ax² + bx + c is a smooth, symmetric curve called a parabola. The shape depends on the sign of a.
二次函数 y = ax² + bx + c 的图像是一条平滑、对称的曲线，称为抛物线。其形状取决于 a 的符号。
- If a > 0, the parabola opens upwards (u-shaped) and has a minimum point.
- If a < 0, the parabola opens downwards (n-shaped) and has a maximum point.
- 若 a > 0，抛物线开口向上（U 形），有最小值点。
- 若 a < 0，抛物线开口向下（n 形），有最大值点。
The larger the absolute value of a, the narrower the parabola. For a = 0, the graph degenerates into a straight line.
a 的绝对值越大，抛物线越窄。当 a = 0 时，图像退化为一条直线。
7. Turning Point and Axis of Symmetry | 顶点与对称轴
Every parabola has a vertical line of symmetry that passes through the turning point (vertex). The x-coordinate of the vertex is given by:
每条抛物线都有一条垂直对称轴，它经过顶点（转向点）。顶点的 x 坐标由以下公式给出：
x = -b / (2a)
To find the y-coordinate of the vertex, substitute this x-value back into the original function.
要求顶点的 y 坐标，将此 x 值代回原函数即可。
Alternatively, if the quadratic is written in completed square form y = a(x - h)² + k, then the vertex is simply (h, k).
或者，如果二次函数写成配方法形式 y = a(x - h)² + k，则顶点坐标直接为 (h, k)。
For example, for y = 2x² - 8x + 5, x = 8/(2×2) = 2. Substituting x = 2 gives y = 2(2)² - 8(2) + 5 = 8 - 16 + 5 = -3. So the vertex is (2, -3).
例如，对于 y = 2x² - 8x + 5，x = 8/(2×2) = 2。代入 x = 2 得 y = 2(2)² - 8(2) + 5 = 8 - 16 + 5 = -3。所以顶点为 (2, -3)。
8. Roots and Intercepts | 根与截距
The roots of a quadratic equation ax² + bx + c = 0 are the x-values where the graph crosses the x-axis. These correspond to the points (r₁, 0) and (r₂, 0).
二次方程 ax² + bx + c = 0 的根是图像与 x 轴交点的 x 坐标。这些交点对应于点 (r₁, 0) 和 (r₂, 0)。
The y-intercept is always at (0, c), because setting x = 0 gives y = c. For example, in y = x² - 3x + 2, the y-intercept is (0, 2).
y 截距总是在 (0, c)，因为令 x = 0 得 y = c。例如，在 y = x² - 3x + 2 中，y 截距为 (0, 2)。
If the discriminant is positive, there are two distinct x-intercepts. If it is zero, the parabola touches the x-axis at exactly one point (the vertex). If negative, the graph never touches the x-axis.
如果判别式为正，则有两个不同的 x 截距。如果为零，抛物线与 x 轴恰好相切于一点（即顶点）。如果为负，则图像与 x 轴无交点。
9. Sketching Quadratics | 画二次函数草图
To sketch a quadratic graph efficiently, use the following step-by-step approach:
要高效地画出二次函数草图，请遵循以下步骤：
- Determine the direction of the parabola from the sign of a.
- Find the y-intercept by substituting x = 0.
- Find the roots by solving the equation ax² + bx + c = 0.
- Find the vertex using x = -b/(2a) and substitute back.
- Plot the points and draw a smooth curve through them.
- 根据 a 的符号确定抛物线的开口方向。
- 令 x = 0 求 y 截距。
- 解方程 ax² + bx + c = 0 求根。
- 使用 x = -b/(2a) 并代回求顶点坐标。
- 描点并通过这些点画一条平滑曲线。
For example, sketch y = -x² + 4x - 3. Here a = -1, so the parabola opens downwards. The y-intercept is (0, -3). Solving -x² + 4x - 3 = 0 gives x = 1 and x = 3. The vertex has x = -4/(2×-1) = 2, and y = -4 + 8 - 3 = 1. So the vertex is (2, 1).
例如，画出 y = -x² + 4x - 3 的草图。这里 a = -1，所以抛物线开口向下。y 截距为 (0, -3)。解 -x² + 4x - 3 = 0 得 x = 1 和 x = 3。顶点的 x = -4/(2×-1) = 2，y = -4 + 8 - 3 = 1。因此顶点为 (2, 1)。
10. Applications in Word Problems | 应用题
Quadratic equations frequently appear in real-world contexts such as projectile motion, area problems, and optimisation problems. The key is to translate the words into a quadratic equation correctly.
二次方程经常出现在现实情境中，例如抛体运动、面积问题和最优化问题。关键是将文字正确地转化为二次方程。
Consider a rectangle whose length is 3 metres more than its width. If the area is 40 square metres, let the width be x. Then the length is x + 3, and the area equation is x(x + 3) = 40.
考虑一个长方形，其长比宽多 3 米。如果面积为 40 平方米，设宽为 x。则长为 x + 3，面积方程为 x(x + 3) = 40。
x² + 3x - 40 = 0
Factorising gives (x + 8)(x - 5) = 0, so x = -8 or x = 5. Since a width cannot be negative, the width is 5 metres and the length is 8 metres.
因式分解得 (x + 8)(x - 5) = 0，所以 x = -8 或 x = 5。因为宽度不可能为负数，所以宽度为 5 米，长度为 8 米。
Always check the feasibility of your solutions within the context of the problem. Discard any root that does not make physical sense.
始终检查你的解在问题情境中是否合理。丢弃任何没有实际意义的根。
11. Common Mistakes | 常见错误
Even strong students often lose marks on quadratic questions due to avoidable errors. Here are some mistakes to watch out for:
即使是优秀的学生，也常常因为一些可以避免的错误而在二次函数题目上丢分。以下是一些需要警惕的错误：
- Forgetting to rearrange the equation into the form ax² + bx + c = 0 before solving.
- Losing negative signs when substituting into the quadratic formula.
- Misinterpreting the discriminant: "no real roots" is not the same as "no roots" in the context of complex numbers.
- Drawing the parabola incorrectly when a < 0 – the vertex must be a maximum.
- Using the wrong x-coordinate for the vertex: remember x = -b/(2a), not b/(2a).
- 在求解前忘记将方程整理为 ax² + bx + c = 0 的形式。
- 代入二次公式时丢失负号。
- 误解读判别式：在复数背景下，“没有实数根”不等于“没有根”。
- 当 a < 0 时画错抛物线——顶点必须是最大值点。
- 使用错误的顶点 x 坐标：记住是 x = -b/(2a)，而不是 b/(2a)。
By being mindful of these pitfalls, you can significantly improve your accuracy in exams.
通过注意这些陷阱，你可以显著提高考试中的准确性。
12. Practice Questions | 练习题
To consolidate your understanding, try the following questions before checking the answers:
为了巩固你的理解，请在查看答案之前尝试以下问题：
- Solve 2x² - 9x + 4 = 0 by factorisation.
- Use completing the square to solve x² + 8x + 2 = 0. Give your answer in surd form.
- Find the discriminant of x² - 6x + 9. What does this tell you about the roots?
- Sketch the graph of y = x² - 2x - 3, labelling the intercepts and the turning point.
- A company makes a profit P(x) = -2x² + 12x - 7, where x is the number of items sold in hundreds. Find the maximum profit and the number of items sold at that profit.
- 用因式分解法解 2x² - 9x + 4 = 0。
- 用配方法解 x² + 8x + 2 = 0，答案以根式表示。
- 求 x² - 6x + 9 的判别式。这告诉你什么关于根的信息？
- 画出 y = x² - 2x - 3 的草图，标出截距和顶点坐标。
- 一家公司的利润为 P(x) = -2x² + 12x - 7，其中 x 是以百件计的销售数量。求最大利润及此时售出的件数。
Answers: 1) x = 1/2 or x = 4. 2) x = -4 ± √14. 3) Discriminant = 0, so there is one repeated root. 4) Vertex (1, -4), roots x = -1 and x = 3, y-intercept (0, -3). 5) Maximum profit is 11,000 (when x = 3, so 300 items).
答案：1) x = 1/2 或 x = 4。2) x = -4 ± √14。3) 判别式为 0，所以有一个重根。4) 顶点 (1, -4)，根 x = -1 和 x = 3，y 截距 (0, -3)。5) 最大利润为 11,000（此时 x = 3，即 300 件）。
If you completed all these correctly, you have a strong foundation for quadratic equations in your IGCSE exam.
如果你全部答对，说明你已经为 IGCSE 考试中的二次方程打下了坚实的基础。
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