📚 IGCSE Mathematics: Mastering Quadratic Equations | IGCSE 数学：掌握二次方程
Quadratic equations appear throughout IGCSE Mathematics, from basic factorising and formula methods to graph sketching and real-life modelling. This revision guide covers standard form, three core solving techniques, the discriminant, graphical interpretation, and common pitfalls that students must avoid in exams.
二次方程贯穿 IGCSE 数学，从基础因式分解、公式法到图像绘制和实际建模都会出现。本复习指南涵盖标准形式、三种核心求解方法、判别式、图像解释以及考试中必须避免的常见错误。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation in which the highest power of the unknown variable is 2. It can always be written in the general form shown below, where a, b and c are constants and a cannot be zero.
二次方程是未知数最高次数为 2 的多项式方程。它总是可以写成如下一般形式，其中 a、b、c 是常数，且 a 不能为零。
ax² + bx + c = 0, a ≠ 0
dition a ≠ 0 is essential. Examples include 2x² + 3x - 5 = 0, x² - 9 = 0 and -x² + 4x = 0.
如果 a = 0，二次项就消失了，方程变成一次方程而不是二次方程，因此条件 a ≠ 0 很关键。常见的例子包括 2x² + 3x - 5 = 0、x² - 9 = 0 和 -x² + 4x = 0。
2. Standard Form and Coefficients | 标准形式与系数
Before solving, always rewrite the equation so that all terms are on one side and the other side is equal to 0. The coefficient of x² is a, the coefficient of x is b, and the constant term is c. This may require expanding brackets, collecting like terms, or clearing fractions first.
求解前，始终先把方程整理为所有项都在一边、另一边等于 0 的形式。x² 的系数是 a，x 的系数是 b，常数项是 c。这可能需要先展开括号、合并同类项或去分母。
The table below shows how to identify a, b and c correctly, including negative signs. A common error is to misread the sign of b or c, especially when moving terms across the equals sign.
下表展示了如何正确识别 a、b、c，包括负号。一个常见错误是误读 b 或 c 的符号，尤其是在跨等号移项时。
| Equation | a | b | c |
| x² - 5x + 6 = 0 | 1 | -5 | 6 |
| 3x² + 2x - 1 = 0 | 3 | 2 | -1 |
| -x² + 4x = 0 | -1 | 4 | 0 |
Once the equation is in standard form, you can choose which solving method is most suitable for the given numbers.
一旦方程写成标准形式，你就可以根据具体数字选择最合适的求解方法。
3. Solving by Factorising | 因式分解法求解
Factorising is often the fastest method when the quadratic expression can be written as a product of two linear factors. The key principle is the zero-product property: if two factors multiply to give zero, then at least one of them must be zero.
当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。其核心原理是零乘积性质：如果两个因式相乘等于零，那么其中至少有一个因式必须为零。
If pq = 0, then p = 0 or q = 0
write both solutions unless the question asks for one valid value in a real-world context.
例如，求解 x² - 5x + 6 = 0。将其因式分解为 (x - 2)(x - 3) = 0。然后令每个括号等于零：x - 2 = 0 得到 x = 2，x - 3 = 0 得到 x = 3。除非题目要求在实际情景中只取一个有效值，否则两个解都要写出。
0 or x = -3. Never divide both sides by x without considering the possibility that x = 0.
因式分解时，先检查是否有公因式，例如 2x² + 6x = 0 中的公因式 2。分解为 2x(x + 3) = 0，得到 x = 0 或 x = -3。切勿两边同除以 x 而不考虑 x = 0 的可能性。
4. The Quadratic Formula | 求根公式
The quadratic formula works for every quadratic equation, including cases where factorising is difficult or impossible. It is especially useful when roots are irrational or when the coefficients are not integers.
求根公式适用于所有二次方程，包括因式分解困难或无法分解的情况。当根是无理数或系数不是整数时，它尤其有用。
x = (-b ± √b² - 4ac) ÷ (2a)
To use the formula, substitute the values of a, b and c very carefully, including any negative signs. Compute the expression under the square root first, then separate the two cases using + and -, and finally divide by 2a.
使用公式时，要非常仔细地代入 a、b、c 的值，包括任何负号。先计算根号下的式子，然后用 + 和 - 分成两种情况，最后除以 2a。
a = 2, b = 3 and c = -2. The discriminant is 3² - 4 × 2 × (-2) = 9 + 16 = 25, so x = (-3 ± √25) ÷ (2 × 2) = (-3 ± 5) ÷ 4. This gives x = 2 ÷ 4 = 1/2 or x = -8 ÷ 4 = -2.
对于 2x² + 3x - 2 = 0，我们有 a = 2、b = 3、c = -2。判别式为 3² - 4 × 2 × (-2) = 9 + 16 = 25，所以 x = (-3 ± √25) ÷ (2 × 2) = (-3 ± 5) ÷ 4。由此得到 x = 2 ÷ 4 = 1/2 或 x = -8 ÷ 4 = -2。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic expression in the form p(x + q)² + r. This method is useful for deducing the turning point of a parabola, solving equations, and proving results in algebra.
配方法将二次式重写为 p(x + q)² + r 的形式。该方法有助于推导抛物线的顶点、求解方程以及在代数中证明结论。
0, so (x + 3)² - 8 = 0. Therefore (x + 3)² = 8, and taking the square root gives x + 3 = ±√8. Since √8 = 2√2, the solutions are x = -3 ± 2√2.
对于 x² + 6x + 1 = 0，取 x 系数的一半，即 3，将 x² + 6x 写成 (x + 3)² - 9。原式变为 (x + 3)² - 9 + 1 = 0，所以 (x + 3)² - 8 = 0。因此 (x + 3)² = 8，开方得到 x + 3 = ±√8。由于 √8 = 2√2，解为 x = -3 ± 2√2。
is method links directly to the vertex form y = a(x - h)² + k, where (h, k) is the vertex.
当 x² 的系数不是 1 时，先从前两项中提出该系数，再配方。该方法直接关联顶点式 y = a(x - h)² + k，其中 (h, k) 是顶点。
6. Graphical Interpretation | 图像解释
. If a is positive, the parabola opens upwards and has a minimum point. If a is negative, it opens downwards and has a maximum point. The y-intercept is always at (0, c).
y = ax² + bx + c 的图像是一条抛物线。如果 a 为正，抛物线开口向上，有一个最低点；如果 a 为负，抛物线开口向下，有一个最高点。y 轴截距总是在 (0, c)。
la can cross the x-axis twice, touch it once, or not cross it at all, depending on the discriminant.
方程 ax² + bx + c = 0 的解就是图像与 x 轴的交点，因为这些点满足 y = 0。根据判别式的不同，抛物线可能与 x 轴相交两次、相切一次，或完全不相交。
2a). In exam questions, you may be asked to sketch the graph, label the intercepts, or write down the coordinates of the turning point.
对称轴是经过顶点的竖直直线，其方程为 x = -b ÷ (2a)。在考试题中，你可能需要画出图像、标出截距，或写出顶点坐标。
7. The Discriminant | 判别式
The discriminant is the expression inside the square root of the quadratic formula. It tells you the nature of the roots without actually solving the equation.
判别式是求根公式中根号内的表达式。它可以在不实际求解方程的情况下判断根的性质。
Discriminant: Δ = b² - 4ac
| Discriminant value | Nature of roots | Graph |
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One repeated real root | Parabola touches x-axis once |
| Δ < 0 | No real roots | Parabola does not cross x-axis |
IGCSE questions often ask you to find the range of values of k for which a quadratic has two distinct real roots, one repeated root, or no real roots. Set up the discriminant inequality and solve it carefully.
IGCSE 题目经常要求你求出使二次方程有两个不同实根、一个重根或没有实根的 k 值范围。需要建立判别式不等式并仔细求解。
8. Word Problems and Modelling | 应用题与建模
Many real-life problems lead to quadratic equations, including area of rectangles, projectile height, product of consecutive integers, and profit problems. Start by defining the unknown with a variable, then build an equation from the given information.
许多实际问题都会引出二次方程，包括矩形面积、抛射高度、连续整数乘积和利润问题。首先用变量定义未知量，然后根据已知信息建立方程。
ive integers is 56. Let the smaller integer be n, so the next is n + 1. Then n(n + 1) = 56
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