📚 Solving Quadratic Equations | 一元二次方程求解
tants and a ≠ 0. Solving such equations is an essential skill for topics as varied as coordinate geometry, functions and problem solving.
在 Edexcel IGCSE 数学考纲中，一元二次方程无处不在。二次方程是指可以写成 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。解这类方程是学习坐标几何、函数和实际应用题等许多内容的关键技能。
1. What is a Quadratic Equation? | 什么是二次方程？
l form is ax² + bx + c = 0. Here are some examples: x² - 5x + 6 = 0, 2x² = 8, and x² = 3x.
二次方程是最高次数为 2 的方程，一般形式为 ax² + bx + c = 0。例如：x² - 5x + 6 = 0、2x² = 8 和 x² = 3x。
s linear, not quadratic. In the form above, b and c can be zero, but the x² term must exist.
需要注意 a 不能为 0。如果 a = 0，方程就变成一次方程，而不是二次方程。在上述形式中，b 和 c 都可以为 0，但 x² 项必须存在。
2. Expanding and Simplifying | 展开与化简
Before solving, quadratic expressions often need to be expanded. For example, (x + 3)(x - 2) is expanded as x² + x - 6. This skill is the reverse of factorisation and is used to check answers.
在求解之前，通常需要先将二次式展开。例如，(x + 3)(x - 2) 展开后得到 x² + x - 6。展开是因式分解的逆运算，也用于检验答案。
The distributive law is central to expansion. Multiply each term in the first bracket by each term in the second bracket. Then collect like terms.
展开的核心是乘法分配律。用第一个括号中的每一项乘以第二个括号中的每一项，然后合并同类项。
(x + p)(x + q) = x² + (p + q)x + pq
3. Factorising Quadratic Expressions | 二次式的因式分解
Factorising means writing a quadratic expression as a product of two binomial brackets. For x² + bx + c, we look for two numbers whose product is c and whose sum is b.
因式分解就是把一个二次式写成两个二项式的乘积。对于 x² + bx + c，我们需要找到两个数，它们的乘积为 c，和为 b。
se 2 × 3 = 6 and 2 + 3 = 5. For x² - 4, use the difference of two squares: (x - 2)(x + 2).
例如，x² + 5x + 6 = (x + 2)(x + 3)，因为 2 × 3 = 6，且 2 + 3 = 5。对于 x² - 4，可以使用平方差公式：(x - 2)(x + 2)。
When a ≠ 1, for example 2x² + 7x + 3, factorisation requires more care. Use methods such as grouping: split the middle term and factor by pairs.
当 a ≠ 1 时，例如 2x² + 7x + 3，因式分解需要更加细心。可以使用分组法：拆开中间项，再分组提取公因式。
4. Solving by Factorisation | 因式分解法解方程
equation factorises, use the zero product property: if AB = 0, then either A = 0 or B = 0.
一旦二次方程能够因式分解，就可以使用零积性质：如果 AB = 0，那么必有 A = 0 或 B = 0。
First factorise: (x - 2)(x - 3) = 0. Therefore x - 2 = 0 or x - 3 = 0, so x = 2 or x = 3.
解方程 x² - 5x + 6 = 0。首先因式分解得到 (x - 2)(x - 3) = 0。因此 x - 2 = 0 或 x - 3 = 0，所以 x = 2 或 x = 3。
ising. For example, x² = 3x becomes x² - 3x = 0, then x(x - 3) = 0, giving x = 0 or x = 3.
在因式分解前，一定要把方程整理为一边为 0 的形式。例如 x² = 3x 应先改写为 x² - 3x = 0，然后 x(x - 3) = 0，得到 x = 0 或 x = 3。
5. Solving by Completing the Square | 配方法解方程
(x + m)² + n. For x² + 6x, since (x + 3)² = x² + 6x + 9, we write x² + 6x = (x + 3)² - 9.
配方法将 x² + bx + c 改写为 (x + m)² + n。对于 x² + 6x，因为 (x + 3)² = x² + 6x + 9，所以 x² + 6x = (x + 3)² - 9。
0 by writing (x + 3)² - 16 = 0. Then (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = -7.
当因式分解困难时，配方法非常有用。例如，解 x² + 6x - 7 = 0，可写成 (x + 3)² - 16 = 0。于是 (x + 3)² = 16，所以 x + 3 = ±4，得到 x = 1 或 x = -7。
The general rule for x² + bx is to add and subtract (b/2)². This is also the foundation of the quadratic formula.
x² + bx 配方的一般规则是加减 (b/2)²。这也是二次公式的推导基础。
6. The Quadratic Formula | 二次公式
bx + c = 0. It is particularly useful when factorisation is impossible or time-consuming.
二次公式可以求解任意一元二次方程 ax² + bx + c = 0。当因式分解不可行或耗时较长时，它特别有用。
x = (-b ± √b² - 4ac) / 2a
lve 2x² + 3x - 2 = 0. Here a = 2, b = 3 and c = -2. Substitute into the formula: x = (-3 ± √9 + 16) / 4 = (-3 ± 5) / 4, so x = 1/2 or x = -2.
例如解 2x² + 3x - 2 = 0。此时 a = 2，b = 3，c = -2。代入公式得 x = (-3 ± √9 + 16) / 4 = (-3 ± 5) / 4，因此 x = 1/2 或 x = -2。
Be careful with signs: when c is negative, b² - 4ac becomes b² + 4a|c|. Always write the exact expression first, then simplify.
请特别注意符号：当 c 为负数时，b² - 4ac 会变成 b² + 4a|c|。先把公式表达式完整写出来，再逐步化简。
7. The Discriminant | 判别式
e discriminant. It tells us how many real roots a quadratic equation has, without solving it fully.
表达式 Δ = b² - 4ac 称为判别式。它可以在不完整求解的情况下，告诉我们一元二次方程有几个实数根。
If Δ > 0, there are two distinct real roots.
若 Δ > 0，方程有两个不同的实数根。
If Δ = 0, there is exactly one repeated real root.
若 Δ = 0，方程有一个二重实数根。
If Δ < 0, there are no real roots.
若 Δ < 0，方程没有实数根。
so the equation has no real roots. The graph of y = x² + 4x + 5 does not cross the x-axis.
对于 x² + 4x + 5 = 0，Δ = 16 - 20 = -4，所以方程没有实数根。图像 y = x² + 4x + 5 与 x 轴不相交。
8. Roots and the Graph of a Quadratic | 方程的根与二次函数图像
= 0 are the x-coordinates of the points where the parabola crosses or touches the x-axis.
对于二次函数 y = ax² + bx + c，方程 ax² + bx + c = 0 的根就是抛物线与 x 轴相交或相切点的横坐标。
he vertex is at x = -b / (2a), and the y-coordinate reflects the minimum or maximum value.
当 a > 0 时，抛物线开口向上；当 a < 0 时，开口向下。顶点横坐标为 x = -b / (2a)，其纵坐标对应函数的最小值或最大值。
uadratic graph. Mark the roots, the y-intercept c, and the axis of symmetry x = -b / (2a).
考试中可能要求画出二次函数草图。请标出根、y 轴截距 c 以及对称轴 x = -b / (2a)。
9. Word Problems | 应用题
× width. If one side is x and the other is x + 3, and the area is 28, then x(x + 3) = 28.
二次方程常出现在几何和数字问题中。例如，矩形面积等于长乘以宽。若一边为 x，另一边为 x + 3，面积为 28，则 x(x + 3) = 28。
orise to get (x + 7)(x - 4) = 0, so x = 4 or x = -7. Since length must be positive, x = 4.
展开后得到 x² + 3x - 28 = 0，因式分解为 (x + 7)(x - 4) = 0，所以 x = 4 或 x = -7。由于长度必须为正，因此 x = 4。
Always reject any negative or impossible solutions in context. State your final answer with appropriate units.
在具体情境中，一定要舍去负数或不符合实际的解，并写出带单位的最终答案。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
bx + c = 0 before solving. Check whether a is 1 or not; choose the most efficient method.
在 Edexcel IGCSE 考试中，求解前务必先把方程整理成 ax² + bx + c = 0 的形式。观察 a 是否为 1，选择最高效的方法。
| Common mistake | Correction |
| Dividing both sides by x | Factorise x out instead, to avoid losing the root x = 0 |
| Forgetting the ± sign in the quadratic formula | Always include ± when taking a square root |
| Sign errors when c is negative | Substitute carefully: b² - 4ac with a negative c means plus |
After solving, substitute your answers back into the original equation to check. Show all working: in Edexcel IGCSE, method marks are often as important as final answers.
解完后，把答案代回原方程检验。在 Edexcel IGCSE 中，方法分往往和最终答案分同等重要，所以请写出完整过程。
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