📚 Mastering Quadratic Equations for IGCSE Mathematics | 精通 IGCSE 数学中的二次方程
Quadratic equations are one of the most important algebraic topics in IGCSE Mathematics. They appear in many forms, from simple factorisation to complex word problems, and they build a foundation for more advanced functions and calculus.
二次方程是 IGCSE 数学中最重要的代数主题之一。它们以多种形式出现，从简单的因式分解到复杂的应用题，为更高级的函数和微积分打下基础。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the unknown variable x is 2. Its general form is:
二次方程是二次多项式方程，即未知变量 x 的最高次数为 2。它的一般形式为：
ax² + bx + c = 0
Here a, b and c are constants, and a ≠ 0. If a were 0, the equation would become linear rather than quadratic.
这里 a、b、c 是常数，且 a ≠ 0。如果 a = 0，该方程就变成一次方程而不是二次方程。
2. Standard Form and Coefficient Rules | 标准形式与系数规则
Before solving a quadratic equation, you should rearrange all terms onto one side so that the other side equals 0. This is called standard form.
在解二次方程之前，应把所有项移到等式一边，使另一边等于0。这称为标准形式。
For example, 2x² = 5x - 3 should be rewritten as 2x² - 5x + 3 = 0.
例如，2x² = 5x - 3 应改写为 2x² - 5x + 3 = 0。
Identify a, b and c accurately: a is the coefficient of x², b is the coefficient of x, and c is the constant term.
准确识别 a、b、c：a 是 x² 的系数，b 是 x 的系数，c 是常数项。
3. Solving by Factorisation | 因式分解法
Factorisation works when the quadratic expression can be written as the product of two linear factors. The method relies on the fact that if two numbers multiply to give 0, at least one of them must be 0.
当二次式可以写成两个一次因式的乘积时，可以使用因式分解法。该方法依据的是：如果两个数相乘为0，则其中至少一个必须为0。
d form, factorise into (px + q)(rx + s) = 0, then set each factor equal to zero and solve.
步骤：将二次方程写成标准形式，因式分解为 (px + q)(rx + s) = 0，然后令每个因式等于0并求解。
Example: Solve x² - 5x + 6 = 0. Factorising gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.
例题：解 x² - 5x + 6 = 0。因式分解得 (x - 2)(x - 3) = 0，所以 x = 2 或 x = 3。
4. Solving by Completing the Square | 配方法
Completing the square transforms a quadratic expression into a perfect square plus or minus a constant. This method is especially useful for finding the vertex of a parabola and for cases where factorisation is difficult.
配方法可将二次式转化为一个完全平方加上或减去一个常数。这种方法在求抛物线顶点以及因式分解困难的情况下尤其有用。
For the general form x² + bx + c = 0, write x² + bx as (x + b/2)² - (b/2)².
对于一般形式 x² + bx + c = 0，将 x² + bx 写成 (x + b/2)² - (b/2)²。
x² + 6x + 2 = 0. First, x² + 6x = (x + 3)² - 9, so the equation becomes (x + 3)² - 7 = 0.
例题：解 x² + 6x + 2 = 0。首先，x² + 6x = (x + 3)² - 9，因此方程变为 (x + 3)² - 7 = 0。
Then (x + 3)² = 7, so x + 3 = ±√7, giving x = -3 ± √7.
然后 (x + 3)² = 7，所以 x + 3 = ±√7，得到 x = -3 ± √7。
5. The Quadratic Formula | 求根公式
solves any quadratic equation ax² + bx + c = 0. It is derived from completing the square.
求根公式是解任何二次方程 ax² + bx + c = 0 的通用方法，它由配方法推导而来。
x = (-b ± √b² - 4ac) / (2a)
Use this formula when factorisation is not straightforward, or when the roots are irrational or involve surds.
当因式分解不直接，或根为无理数或带根号时，可使用该公式。
6. The Discriminant and Nature of Roots | 判别式与根的性质
The expression inside the square root, b² - 4ac, is called the discriminant. It tells you how many real solutions a quadratic equation has without actually solving it.
根号内的表达式 b² - 4ac 称为判别式。它可以在不实际求解的情况下告诉你二次方程有多少个实数解。
- If b² - 4ac > 0, there are two distinct real roots.
- 如果 b² - 4ac > 0，则有两个不同的实根。
- If b² - 4ac = 0, there is exactly one real root (a repeated root).
- 如果 b² - 4ac = 0，则恰好有一个实根（重根）。
- If b² - 4ac < 0, there are no real roots; the roots are complex.
- 如果 b² - 4ac < 0，则没有实根；根为复数。
7. Graphical Interpretation: Parabolas and x-Intercepts | 图像解读：抛物线与 x 轴交点
solutions to ax² + bx + c = 0 are the x-coordinates where the parabola crosses the x-axis.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解就是抛物线与 x 轴交点的 x 坐标。
If the discriminant is positive, the parabola crosses the x-axis at two points. If it is zero, the parabola just touches the x-axis at one point. If it is negative, the parabola does not cross the x-axis.
如果判别式为正，抛物线与 x 轴相交于两点；如果为零，抛物线刚好与 x 轴相切于一点；如果为负，抛物线与 x 轴没有交点。
8. Common Mistakes to Avoid | 常见错误避免
Many students forget to set the equation to 0 before factorising. Always rearrange first.
许多学生在因式分解前忘记将方程设为 0。一定要先整理方程。
Another common error is dividing both sides by x when x could be 0. This can lose a solution. Instead, factorise and set each factor to zero.
另一个常见错误是当 x 可能为0时两边同除以 x，这可能丢失一个解。应进行因式分解并令每个因式为0。
Be careful with signs when using the quadratic formula, especially when b is negative and when dealing with -b.
使用求根公式时要注意符号，尤其是当 b 为负以及处理 -b 时。
9. Exam-Style Worked Example | 考试型例题
Question: Solve 2x² - 7x - 15 = 0, giving your answers exactly.
题目：解方程 2x² - 7x - 15 = 0，给出精确答案。
Step 1: Identify a = 2, b = -7, c = -15.
步骤1：确定 a = 2，b = -7，c = -15。
mbers that multiply to give 2 × (-15) = -30 and add to give -7. The numbers are -10 and 3.
步骤2：尝试因式分解。需要两个数，使它们的乘积为 2 × (-15) = -30，和为 -7。这两个数是 -10 和 3。
- 10x + 3x - 15 = 0. Factor by grouping: 2x(x - 5) + 3(x - 5) = 0, so (2x + 3)(x - 5) = 0.
拆分中间项：2x² - 10x + 3x - 15 = 0。分组因式分解：2x(x - 5) + 3(x - 5) = 0，因此 (2x + 3)(x - 5) = 0。
Step 3: Set each factor to zero: 2x + 3 = 0 gives x = -3/2; x - 5 = 0 gives x = 5.
步骤3：令每个因式为0：2x + 3 = 0 得 x = -3/2；x - 5 = 0 得 x = 5。
Final answer: x = -3/2 or x = 5.
最终答案：x = -3/2 或 x = 5。
10. Summary Checklist | 总结清单
Before an IGCSE exam, make sure you can complete these tasks confidently:
在 IGCSE 考试前，确保你能够自信地完成以下任务：
- Rearrange a quadratic into standard form ax² + bx + c = 0.
- 将二次方程整理成标准形式 ax² + bx + c = 0。
- Factorise simple and harder quadratics, including cases where a ≠ 1.
- 对简单和较难的二次式进行因式分解，包括 a ≠ 1 的情况。
- Complete the square and use the result to solve equations or find turning points.
- 配方法并用结果解方程或求顶点。
- Apply the quadratic formula accurately with attention to signs.
- 准确应用求根公式，注意符号。
- Use the discriminant to comment on the number and nature of roots.
- 使用判别式判断根的个数和性质。
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