📚 Solving Quadratic Equations | 解二次方程
A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, and real-life modelling questions, so you need to be confident with several solution methods.
二次方程是 IGCSE 数学中最重要的主题之一。它在代数、图像和实际建模题中都会出现，因此你需要熟练掌握多种求解方法。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation in one variable can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were zero, the equation would become linear.
一元二次方程可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。如果 a 为 0，该方程就变成了一次方程。
For example, 2x² − 3x + 1 = 0 is quadratic, but 3x + 1 = 0 is linear. IGCSE questions usually ask you to solve the equation, sketch the related graph, or interpret the roots in context.
例如，2x² − 3x + 1 = 0 是二次方程，而 3x + 1 = 0 是一次方程。IGCSE 题目通常要求你解方程、画出相关图像或结合实际情境解释根的意义。
2. Solving by Factorising | 因式分解法
When the quadratic expression factorises easily, write it as a product of two brackets: (px + q)(rx + s) = 0. Then use the zero product property: if a product is zero, at least one factor must be zero.
当二次式容易因式分解时，把它写成两个括号的乘积：(px + q)(rx + s) = 0。然后使用零乘积性质：若乘积为 0，则至少有一个因式为 0。
Example: x² − 5x + 6 = 0 becomes (x − 2)(x − 3) = 0, so x = 2 or x = 3. Always check your factors by expanding them back.
例：x² − 5x + 6 = 0 可化为 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。一定要通过展开来检验因式是否正确。
3. Completing the Square | 配方法
Completing the square rewrites x² + bx + c as (x + b/2)² − (b/2)² + c. This method is useful when factorising is difficult, and it also helps find the vertex of a quadratic graph.
配方法将 x² + bx + c 改写为 (x + b/2)² − (b/2)² + c。当因式分解较难时，该方法很有用，同时它也有助于求二次函数图像的顶点。
x² + bx + c = (x + b/2)² − (b/2)² + c
For example, x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7. To solve it, set (x + 3)² = 7 and then take the square root of both sides.
例如，x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。要解方程，可令 (x + 3)² = 7，然后对两边开平方。
4. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is derived by completing the square and is given below. It is especially useful when the coefficient of x² is not 1.
求根公式可以解任何二次方程 ax² + bx + c = 0。它由配方法推导而来，公式如下。当 x² 的系数不是 1 时，公式尤其有用。
x = (−b ± √(b² − 4ac)) / 2a
Substitute a, b and c carefully, especially if b is negative. For example, in 2x² − 4x − 6 = 0, a = 2, b = −4 and c = −6.
代入 a、b、c 时要特别小心，尤其是当 b 为负数时。例如，在 2x² − 4x − 6 = 0 中，a = 2，b = −4，c = −6。
5. Discriminant and Number of Roots | 判别式与根的个数
The discriminant is D = b² − 4ac. It tells you how many real roots the quadratic has: D > 0 gives two distinct real roots, D = 0 gives one repeated real root, and D < 0 gives no real roots.
判别式是 D = b² − 4ac。它可以判断二次方程有多少个实数根：D > 0 有两个不同实根，D = 0 有一个重根，D < 0 没有实数根。
| Discriminant | 判别式 | Number of real roots | 实数根个数 |
|---|---|
| D > 0 | Two distinct real roots | 两个不同实根 |
| D = 0 | One repeated real root | 一个重根 |
| D < 0 | No real roots | 没有实数根 |
This idea is often tested in IGCSE questions that ask how many times a curve crosses the x-axis.
IGCSE 题目经常考察这个概念，通常会问曲线与 x 轴相交多少次。
6. Graphical Interpretation | 图像解释
The solutions of ax² + bx + c = 0 are the x-coordinates where the parabola y = ax² + bx + c crosses the x-axis. A repeated root touches the x-axis at one point, and no real roots mean the graph never crosses the x-axis.
方程 ax² + bx + c = 0 的解就是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。重根表示图像在一点接触 x 轴，没有实根表示图像不与 x 轴相交。
If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
如果 a > 0，抛物线开口向上且有最小值点。如果 a < 0，抛物线开口向下且有最大值点。
7. Choosing the Best Method | 选择最佳方法
In the exam, first try factorising if the numbers look simple. If not, use the quadratic formula or complete the square. The formula is the most reliable method and works for every quadratic equation.
考试时，如果数字简单，先尝试因式分解。如果不行，就使用求根公式或配方法。求根公式是最可靠的方法，适用于所有二次方程。
| Equation type | 方程类型 | Recommended method | 推荐方法 |
|---|---|
| Simple integer roots | 整数根 | Factorising | 因式分解 |
| Messy or surd roots | 无理根 | Quadratic formula or completing the square | 求根公式或配方法 |
| Finding the vertex | 求顶点 | Completing the square | 配方法 |
8. Common Mistakes | 常见错误
One common mistake is forgetting to set the equation to 0 before factorising. For example, x² − 3x = 4 must be written as x² − 3x − 4 = 0 first.
一个常见错误是在因式分解前忘记先把方程化为 0。例如，x² − 3x = 4 必须先写成 x² − 3x − 4 = 0。
Another mistake is dividing both sides by x and losing the solution x = 0. Instead, factorise and keep all possible roots. Sign errors in the quadratic formula are also common when b is negative.
另一个错误是两边同时除以 x，从而丢失 x = 0 这个解。正确做法是因式分解并保留所有可能的根。当 b 为负数时，求根公式中的符号错误也很常见。
9. Word Problems and Modelling | 应用题与建模
A typical IGCSE question gives a real-life situation leading to a quadratic equation, such as area, projectile motion, or profit. Define the unknown, form the equation, solve it, and check that the answer makes sense in the original context.
典型的 IGCSE 题目会给出一个实际情境，如面积、抛体运动或利润，并建立二次方程。设未知数，列方程，求解，并检查答案在原情境中是否合理。
For example, if a rectangle has length x + 3 and width x, with area 40, write x(x + 3) = 40, simplify to x² + 3x − 40 = 0, then solve.
例如，一个矩形的长为 x + 3，宽为 x，面积为 40，则可写出 x(x + 3) = 40，化简为 x² + 3x − 40 = 0，然后求解。
10. Exam Tips and Summary | 考试技巧与总结
Always write the formula down, substitute carefully, and simplify the surd if possible. Show your working because method marks are available even if you make a small arithmetic error.
始终先写出公式，仔细代入，并尽可能化简根式。写出解题步骤，因为即使出现小的算术错误，也能获得方法分。
Finally, check your solutions by substituting them back into the original equation. This helps you catch sign errors and incorrect factorisation quickly.
最后，把解代回原方程进行检验。这可以帮助你快速发现符号错误和因式分解错误。
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