📚 Solving Quadratic Equations for IGCSE Maths | IGCSE 数学二次方程求解
nstants and a ≠ 0. You are expected to solve quadratics by factorising, completing the square, using the quadratic formula, and interpreting graphs. This revision guide covers all the essential methods and common exam pitfalls.
在 IGCSE 数学中，二次方程出现在代数、坐标几何和现实生活建模中。二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。考试要求你掌握因式分解法、配方法、二次公式法以及图像解读法。本复习指南涵盖所有核心方法和常见考试易错点。
1. Standard Form and Key Terms | 标准形式与关键术语
condition a ≠ 0 is essential because if a = 0 the equation becomes linear, not quadratic.
二次方程是指任何可以整理成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为实数，且 a 不等于零。ax² 称为二次项，bx 称为一次项，c 称为常数项。a ≠ 0 这个条件很关键，因为如果 a = 0，方程就变成了一次方程，而不是二次方程。
kes factorising, completing the square, and substituting into the formula much easier and reduces sign errors.
在选择求解方法之前，务必先把方程化为标准形式。例如，x² + 3x = 10 必须先写成 x² + 3x − 10 = 0。这一步会让因式分解、配方以及代入公式更加容易，并能减少符号错误。
2. Solving by Factorisation | 因式分解法
+ 5x + 6 = 0 can be factorised as (x + 2)(x + 3) = 0, giving solutions x = -2 and x = -3.
如果二次表达式 ax² + bx + c 可以写成两个一次因式 (px + q)(rx + s) 的乘积，那么方程就能快速求解。对于 a = 1 的简单二次方程，因式分解通常是最快的方法。例如，x² + 5x + 6 = 0 可以因式分解为 (x + 2)(x + 3) = 0，得到 x = -2 和 x = -3。
x + 3 = 0, then factorising by grouping gives (2x + 1)(x + 3) = 0, so x = -1/2 or x = -3.
当 a 不等于 1 时，因式分解仍然可行，但需要更加仔细。对于 2x² + 7x + 3 = 0，我们寻找两个数，使它们的乘积为 ac = 6，和为 b = 7，这两个数是 6 和 1。拆分中间项得到 2x² + 6x + x + 3 = 0，然后分组分解得到 (2x + 1)(x + 3) = 0，所以 x = -1/2 或 x = -3。
3. The Zero Product Property | 零乘积性质
perty only works when the right-hand side is exactly zero, which is why standard form is so important.
仅因式分解本身不能给出解，还必须使用零乘积性质。如果两个因式的乘积为零，那么至少有一个因式为零。也就是说 (px + q)(rx + s) = 0 意味着 px + q = 0 或 rx + s = 0。这个性质只在右边恰好为零时成立，因此标准形式非常重要。
s by x and lose the root x = 0, so always keep the factors and set each one equal to zero.
例如，如果 x(x − 4) = 0，那么要么 x = 0，要么 x − 4 = 0，得到 x = 0 或 x = 4。一个常见错误是两边同除以 x 而丢失 x = 0 这个根，所以应该保留因式并令每个因式等于零。
4. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This method is especially useful when the quadratic does not factorise neatly, and it also helps find the vertex of a quadratic graph. The core identity for monic quadratics is shown below.
配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式。当二次方程不容易因式分解时，这个方法特别有用，而且它还有助于求二次图像的顶点。首项系数为 1 的二次方程的核心恒等式如下所示。
x² + bx = (x + b/2)² − (b/2)²
or example, x² + 6x + 1 = 0 becomes (x + 3)² − 9 + 1 = 0, so (x + 3)² = 8, giving x = -3 ± √8 u the vertex of y = x² + 6x + 1 is (-3, -8), which is useful in graph sketching questions.
例如，x² + 6x + 1 = 0 变为 (x + 3)² − 9 + 1 = 0，所以 (x + 3)² = 8，得到 x = -3 ± √8。这个形式还能告诉你 y = x² + 6x + 1 的顶点是 (-3, -8)，这在画图题中很有用。
5. The Quadratic Formula | 二次公式
rived by completing the square on the general quadratic and is given below. You must memorise this formula exactly and use brackets carefully when substituting negative values.
二次公式可以求解任何二次方程 ax² + bx + c = 0。它是对一般二次方程配方法推导出来的，公式如下。你必须准确记住这个公式，并在代入负值时仔细使用括号。
x = (-b ± √b² − 4ac) ÷ (2a)
tuting into the formula. Example: for 2x² + 3x − 2 = 0, a = 2, b = 3, c = -2, so x = (-3 ± √9 + 16 or x = -2. This method is especially useful when factorisation is difficult or impossible.
代入公式前，务必先将方程写成标准形式。例如：对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = -2，所以 x = (-3 ± √9 + 16) ÷ 4 = (-3 ± 5) ÷ 4，得到 x = 1/2 或 x = -2。当因式分解困难或不可能时，这个方法特别有用。
6. The Discriminant | 判别式
The expression b² − 4ac inside the square root is called the discriminant. It tells you how many real roots a quadratic equation has without solving it fully. This idea appears in many IGCSE questions, especially those asking for values of k that give equal roots or no real roots.
平方根内的式子 b² − 4ac 称为判别式。它可以在不完整求解的情况下告诉你二次方程有多少个实数根。这个知识点出现在许多 IGCSE 考题中，尤其是求使方程有等根或无实数根的 k 值的题目。
| Discriminant | Nature of roots |
|---|---|
| b² − 4ac > 0 | Two distinct real roots |
| b² − 4ac = 0 | One repeated real root |
| b² − 4ac < 0 | No real roots |
− 36 = 0, so k = ±6. This type of question is very common and usually worth several marks.
例如，要使 x² + kx + 9 = 0 有等根，令判别式等于零：k² − 36 = 0，所以 k = ±6。这类题目非常常见，通常值好几分。
7. Graphical Interpretation | 图像解读
bx + c crosses the x-axis. If the graph touches the x-axis at one point, there is one repeated root; if it does not cross, there are no real roots. This connects algebra with the graph of a parabola.
ax² + bx + c = 0 的解就是图像 y = ax² + bx + c 与 x 轴交点的横坐标。如果图像只与 x 轴接触于一点，则有一个重根；如果不与 x 轴相交，则没有实数根。这把代数与抛物线图像联系了起来。
If a is positive, the parabola opens upwards, so the graph has a minimum point. If a is negative, it opens downwards and has a maximum point. The number of x-axis crossings is exactly the number of real roots, which can also be checked using the discriminant.
如果 a 为正，抛物线开口向上，因此图像有一个最低点。如果 a 为负，抛物线开口向下，图像有一个最高点。与 x 轴交点的个数恰好就是实数根的个数，这也可以用判别式来验证。
8. Solving by Graphing | 图像法求解
ample by setting one side to zero or linking two graphs such as y = x² + 2x and y = x + 5.
考试中可能会给出图像，要求你通过读取交点坐标来解方程。或者，你可能需要将给定方程重新整理以匹配图像，例如令一边为零，或把两个图像如 y = x² + 2x 与 y = x + 5 联系起来。
For example, to solve x² + 2x − 3
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