📚 Solving Quadratic Equations | 解二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, calculator-paper questions, word problems and graph sketching. Mastering how to solve them quickly and accurately is essential to secure top marks.
二次方程是 IGCSE 数学中最重要的话题之一。它们出现在代数、计算器试卷、应用题以及图像作图中。掌握准确、快速地求解二次方程，是取得高分的关键。
1. What is a Quadratic Equation? | 什么是二次方程
A quadratic equation is any equation that can be written in the standard form:
ax² + bx + c = 0, where a ≠ 0
Here, x is the unknown variable, and a, b, c are real numbers with a non-zero. The highest power of x is 2, which is why it is called “quadratic”.
这里的 x 是未知数，a、b、c 是实数，且 a ≠ 0。未知数的最高次数为 2，因此称为“二次”方程。
For example, 3x² − 5x + 2 = 0 and x² = 16 are both quadratic equations, while x³ − 2x = 0 is not.
例如，3x² − 5x + 2 = 0 和 x² = 16 都是二次方程，而 x³ − 2x = 0 不是。
2. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method when the quadratic can be written as a product of two linear brackets. The key idea is that if two numbers multiply to give zero, then one of them must be zero.
因式分解法通常是求解二次方程最快的方法，前提是方程可以写成两个一次括号相乘的形式。关键思想是：如果两个数相乘等于零，那么其中至少有一个为零。
Step 1: Write the equation in the form ax² + bx + c = 0. Then factorise the left-hand side.
第一步：将方程写成 ax² + bx + c = 0 的形式，然后对左侧进行因式分解。
For example, solve x² − 5x + 6 = 0. We need two numbers that multiply to 6 and add to −5. These are −2 and −3.
例如，解 x² − 5x + 6 = 0。我们需要找两个数，相乘得 6，相加得 −5。这两个数是 −2 和 −3。
(x − 2)(x − 3) = 0
Therefore, x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
因此 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。
If the coefficient of x² is not 1, look for a common factor first, or use trial with brackets such as (px + q)(rx + s).
如果 x² 的系数不是 1，先寻找公因数，或者尝试 (px + q)(rx + s) 形式的括号。
3. Special Cases: Difference of Two Squares | 特殊情形：平方差
A common quadratic pattern is the difference of two squares:
x² − a² = (x − a)(x + a)
This pattern appears in questions such as x² − 9 = 0. Since 9 = 3², we can write (x − 3)(x + 3) = 0, giving x = 3 or x = −3.
这个模式常出现在像 x² − 9 = 0 这样的题目中。因为 9 = 3²，我们可以写成 (x − 3)(x + 3) = 0，得到 x = 3 或 x = −3。
Another example: 4x² − 25 = 0. Here 4x² = (2x)² and 25 = 5², so (2x − 5)(2x + 5) = 0, hence x = 5/2 or x = −5/2.
另一个例子：4x² − 25 = 0。这里 4x² = (2x)²，25 = 5²，所以 (2x − 5)(2x + 5) = 0，因此 x = 5/2 或 x = −5/2。
4. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² + q. This method works for any quadratic equation and is especially useful when the equation cannot be factorised easily.
配方法将二次式改写为 (x + p)² + q 的形式。这种方法适用于任何二次方程，尤其适合不易因式分解的方程。
For the equation x² + bx + c = 0, we use:
x² + bx = (x + b/2)² − (b/2)²
Example: solve x² + 6x + 5 = 0. First, x² + 6x = (x + 3)² − 9. So the equation becomes (x + 3)² − 9 + 5 = 0, which simplifies to (x + 3)² − 4 = 0.
例如：解 x² + 6x + 5 = 0。首先 x² + 6x = (x + 3)² − 9。因此原方程变为 (x + 3)² − 9 + 5 = 0，即 (x + 3)² − 4 = 0。
Now take the square root of both sides: x + 3 = ±2. Thus x = −1 or x = −5.
现在两边开平方：x + 3 = ±2。因此 x = −1 或 x = −5。
If the coefficient of x² is greater than 1, divide the whole equation by that coefficient first, or factor it out before completing the square.
如果 x² 的系数大于 1，先除以这个系数，或者先将其提出，再进行配方。
5. The Quadratic Formula | 二次公式
The quadratic formula is a universal method. It gives the exact solutions of any quadratic equation ax² + bx + c = 0.
二次公式是一种通用方法。它给出了任意二次方程 ax² + bx + c = 0 的精确解。
x = (−b ± √(b² − 4ac)) / (2a)
To use it, substitute the values of a, b and c carefully. For example, solve 2x² − 3x − 2 = 0. Here a = 2, b = −3, c = −2.
使用时，仔细代入 a、b、c 的值。例如，解 2x² − 3x − 2 = 0。这里 a = 2，b = −3，c = −2。
x = (3 ± √((−3)² − 4 × 2 × (−2))) / (2 × 2) = (3 ± √(9 + 16)) / 4 = (3 ± 5) / 4
So x = (3 + 5)/4 = 2 or x = (3 − 5)/4 = −1/2.
所以 x = (3 + 5)/4 = 2 或 x = (3 − 5)/4 = −1/2。
Always show the substitution clearly in exam, as marks are often awarded for each step.
在考试中，一定要清晰地写出代入过程，因为每个步骤都可能得分。
6. The Discriminant | 判别式
The discriminant is the part of the quadratic formula under the square root: Δ = b² − 4ac. It tells us how many real roots a quadratic equation has without solving it fully.
判别式是二次公式中根号内的部分：Δ = b² − 4ac。它无需完全解方程，即可告诉我们二次方程有多少个实数根。
| Discriminant Δ | Number of real roots | 中文说明 |
|---|---|---|
| Δ > 0 | Two distinct real roots | 两个不同实数根 |
| Δ = 0 | One repeated real root | 一个重根（两个相等实数根） |
| Δ < 0 | No real roots | 没有实数根 |
For example, the equation x² + 2x + 5 = 0 has Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16, so it has no real roots.
例如，方程 x² + 2x + 5 = 0 的 Δ = 2² − 4 × 1 × 5 = 4 − 20 = −16，因此它没有实数根。
7. Solving Word Problems | 应用题
Many exam questions require you to form a quadratic equation from a real-life situation. Study the problem carefully, define the unknown variable, and set up the equation before solving.
许多考试题目要求你从实际情境中建立二次方程。仔细审题，定义未知数，并先建立方程，然后再求解。
Example: The length of a rectangle is 3 cm more than its width. The area is 40 cm². Find the width.
例如：一个长方形的长比宽多 3 cm，面积为 40 cm²。求宽。
Let the width be x cm. Then the length is (x + 3) cm. Area = x(x + 3) = 40, so x² + 3x − 40 = 0.
设宽为 x cm，则长为 (x + 3) cm。面积 = x(x + 3) = 40，因此 x² + 3x − 40 = 0。
Factorise: (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a width cannot be negative, the width is 5 cm.
因式分解：(x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于宽不能为负数，所以宽为 5 cm。
Always check whether both solutions make sense in the context of the problem.
始终检查两个解在问题情境中是否都有意义。
8. Graphs and Quadratic Equations | 图像与二次方程
The solutions of a quadratic equation correspond to the x-coordinates where the graph of y = ax² + bx + c crosses the x-axis. These points are called roots or x-intercepts.
二次方程的解对应二次函数 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。这些点称为根或 x 轴截距。
For example, the equation x² − 4x + 3 = 0 can be solved by factorising to get x = 1 and x = 3. On the graph, the parabola crosses the x-axis at (1, 0) and (3, 0).
例如，x² − 4x + 3 = 0 通过因式分解可得到 x = 1 和 x = 3。在图像上，抛物线在 (1, 0) 和 (3, 0) 处穿过 x 轴。
If the discriminant is negative, the graph does not touch the x-axis at all.
如果判别式为负，图像根本不会接触 x 轴。
When a quadratic equation has the form x² = k, its graph intersects the x-axis at two symmetrical points x = √k and x = −√k.
当二次方程为 x² = k 的形式时，其图像在 x 轴上相交于两个对称点 x = √k 和 x = −√k。
9. Common Mistakes | 常见错误
Many students lose marks in quadratic questions due to avoidable mistakes. Pay attention to the following pitfalls.
许多学生在二次方程题目中因可以避免的错误而失分。请注意以下常见陷阱。
First, forgetting to rearrange the equation into standard form before solving. For example, x² = 3x cannot be solved by dividing both sides by x, because x may be zero. The correct way is x² − 3x = 0, then x(x − 3) = 0.
第一，忘记将方程整理成标准形式。例如，x² = 3x 不能直接两边除以 x，因为 x 可能为零。正确方法是 x² − 3x = 0，然后 x(x − 3) = 0。
Second, forgetting to include the ± sign when taking square roots. If x² = 16, then x = ±4, not just x = 4.
第二，开平方时忘记 ± 号。如果 x² = 16，那么 x = ±4，而不仅仅是 x = 4。
Third, using an incorrect factorisation. Always expand your brackets to check whether they multiply back to the original quadratic.
第三，因式分解错误。始终展开括号，检验是否乘回原二次式。
Fourth, when using the quadratic formula, getting the sign of b wrong. For −b, if b is negative, then −b becomes positive.
第四，使用二次公式时把 b 的符号弄错。对于 −b，如果 b 是负数，那么 −b 就是正数。
10. Exam Tips | 考试技巧
In the IGCSE exam, always read the question to see whether it asks for exact answers or decimal approximations. If no method is specified, factorisation is often the quickest when possible.
在 IGCSE 考试中，务必看清题目要求精确答案还是近似小数。如果未指定方法，在可行时因式分解通常最快。
Before starting, try to check if the discriminant is a perfect square. If so, the quadratic can be factorised with rational numbers.
开始前，可以检查判别式是否为完全平方数。如果是，则二次式可以用有理数因式分解。
Show all working clearly. Even if your final answer is wrong, you may still earn method marks for correct substitution or factorisation steps.
清晰展示所有过程。即使最终答案错误，你仍可能因正确的代入或因式分解步骤获得方法分。
Always check your answers by substituting the roots back into the original equation. This simple step catches most arithmetic errors.
始终通过将根代回原方程来检验答案。这个简单的步骤能发现大多数计算错误。
When solving word problems, write a concluding sentence with the correct units. This not only earns communication marks but also shows the examiner your answer is meaningful.
在解应用题时，写出带有正确单位的结论句。这不仅能获得沟通分，还能向考官展示你的答案是合理的。
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