📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程精讲
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, and real-world problems, so understanding how to form, solve, and interpret them is essential for exam success.
二次方程是 IGCSE 数学中最重要的主题之一。它们出现在代数、图像和实际问题中，因此掌握如何建立、求解和解释二次方程对考试成功至关重要。
1. What Is a Quadratic Equation? | 什么是二次方程
A quadratic equation is an equation where the highest power of the variable, usually x, is 2. Its general form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
二次方程是变量的最高次数为 2 的方程，通常用 x 表示。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。
ax² + bx + c = 0 (a ≠ 0)
If a = 0, the equation becomes linear, so a must not equal zero for the equation to remain quadratic.
如果 a = 0，方程就变成一次方程，因此 a 必须不等于零才能保持为二次方程。
2. Standard Form and Key Terms | 标准形式与关键概念
Always rearrange a quadratic equation into the form ax² + bx + c = 0 before solving. This makes the coefficients a, b, and c clear for factorisation, completing the square, or the quadratic formula.
在求解之前，始终将二次方程整理为 ax² + bx + c = 0 的形式。这样做能清楚地看出系数 a、b、c，便于因式分解、配方法或使用求根公式。
The term ax² is the quadratic term, bx is the linear term, and c is the constant term. In IGCSE questions, you may need to expand brackets or collect like terms first to reach the standard form.
ax² 是二次项，bx 是一次项，c 是常数项。在 IGCSE 题目中，你可能需要先展开括号或合并同类项，才能得到标准形式。
3. Solving by Factorisation | 因式分解法
Factorisation works when the quadratic can be written as a product of two linear factors. For example, x² + 5x + 6 = 0 factorises as (x + 2)(x + 3) = 0.
当二次方程可以写成两个一次因式的乘积时，可以用因式分解法。例如 x² + 5x + 6 = 0 因式分解为 (x + 2)(x + 3) = 0。
Setting each factor equal to zero gives x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.
令每个因式等于零，得到 x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。
The key steps are: write the equation in standard form, factorise the quadratic expression, set each factor equal to 0, then solve each linear equation.
关键步骤是：将方程写成标准形式，对二次式进行因式分解，令每个因式等于 0，然后解每个一次方程。
4. Difference of Two Squares | 平方差公式
The difference of two squares is a common special case: a² − b² = (a + b)(a − b). For example, x² − 9 = 0 factorises as (x + 3)(x − 3) = 0, giving x = ±3.
平方差是一个常见的特殊情况：a² − b² = (a + b)(a − b)。例如 x² − 9 = 0 因式分解为 (x + 3)(x − 3) = 0，解得 x = ±3。
Always check whether the constant term is a perfect square and whether the quadratic has no x term. Expressions like 4x² − 25 can be factorised as (2x + 5)(2x − 5).
始终检查常数项是否为完全平方数，以及二次方程是否缺少一次项。像 4x² − 25 这样的式子可以因式分解为 (2x + 5)(2x − 5)。
5. Completing the Square | 配方法
Completing the square transforms a quadratic into the form (x + p)² + q = 0. For x² + 6x + 2 = 0, we write (x + 3)² − 7 = 0, because (x + 3)² = x² + 6x + 9, so we subtract 7.
配方法将二次方程转化为 (x + p)² + q = 0 的形式。对于 x² + 6x + 2 = 0，我们写成 (x + 3)² − 7 = 0，因为 (x + 3)² = x² + 6x + 9，所以需要减去 7。
Then solve: (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7.
然后求解：(x + 3)² = 7，因此 x + 3 = ±√7，得到 x = −3 ± √7。
This method is especially useful when the quadratic does not factorise neatly, and it is often tested in IGCSE questions that ask for exact surd answers.
当二次方程不容易因式分解时，这种方法尤其有用。IGCSE 题目中经常考查配方法并要求给出带有根号的精确答案。
6. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation ax² + bx + c = 0:
求根公式可以求解任何二次方程 ax² + bx + c = 0：
x = (−b ± √(b² − 4ac)) / 2a
Substitute the values of a, b, and c carefully, including negative signs. For 2x² + 3x − 2 = 0, a = 2, b = 3, and c = −2.
代入 a、b、c 的值时要仔细，包括负号。对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2。
Then x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4, giving x = ½ or x = −2.
然后 x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4，解得 x = ½ 或 x = −2。
7. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant is the expression under the square root in the quadratic formula: D = b² − 4ac.
判别式是求根公式中平方根下的表达式：D = b² − 4ac。
If D > 0, there are two distinct real roots. If D = 0, there is exactly one repeated real root. If D < 0, there are no real roots.
如果 D > 0，方程有两个不相等的实根；如果 D = 0，方程有一个重根；如果 D < 0，方程没有实根。
| D = b² − 4ac | Number of real roots |
| D > 0 | 2 distinct real roots |
| D = 0 | 1 repeated real root |
| D < 0 | 0 real roots |
8. Graphical Interpretation | 图像解释
The graph of y = ax² + bx + c is a parabola. The solutions of ax² + bx + c = 0 are the x-intercepts 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 轴交点的横坐标。
If the discriminant is positive, the parabola crosses the x-axis twice. If it is zero, the vertex touches the x-axis. If negative, the parabola does not cross the x-axis.
如果判别式为正，抛物线与 x 轴相交两次；如果为零，顶点与 x 轴相切；如果为负，抛物线不与 x 轴相交。
For positive a, the parabola opens upwards; for negative a, it opens downwards.
当 a 为正时，抛物线开口向上；当 a 为负时，抛物线开口向下。
9. Common Mistakes | 常见错误
One common mistake is forgetting to set the equation to 0 before factorising. Another is dividing both sides by x, which may lose the root x = 0.
一个常见错误是在因式分解前忘记将方程整理为等于 0。另一个错误是两边同时除以 x，这可能会丢失 x = 0 这个根。
Also, watch negative signs when using the quadratic formula, especially when c is negative. A small sign error can change the entire answer.
此外，使用求根公式时要注意负号，尤其是当 c 为负数时。一个小的符号错误可能会改变整个答案。
Always check your solutions by substituting them back into the original equation.
始终通过将解代回原方程来检查答案。
10. Exam Tips | 考试技巧
In IGCSE exams, show your method clearly, even if you use a calculator. Factorise first if possible; otherwise use the quadratic formula or completing the square.
在 IGCSE 考试中，即使使用计算器，也要清晰地展示解题步骤。首先尝试因式分解；如果不行，再使用求根公式或配方法。
Leave answers in exact form unless the question asks you to round. For word problems, check whether a negative solution makes sense in the real-world context.
除非题目要求四舍五入，否则保留精确答案。对于应用题，要检查负解在实际问题中是否合理。
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