📚 Solving Quadratic Equations for IGCSE | IGCSE 数学：解二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, and real-life problems. This revision guide explains the main methods: factorising, completing the square, and the quadratic formula, with tips to avoid common mistakes.
二次方程是 IGCSE 数学中最重要的主题之一，出现在代数、图像和现实问题中。本复习指南讲解主要方法：因式分解、配方法和二次公式，并给出避免常见错误的技巧。
1. What Is a Quadratic Equation? | 什么是二次方程？
are constants and a ≠ 0. The highest power of x is 2, so every quadratic equation has at most two solutions, called roots.
二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。x 的最高次数是 2，因此每个二次方程最多有两个解，称为根。
b = −5 and c = 3. If the coefficient of x² were zero, the equation would not be quadratic.
例如，2x² − 5x + 3 = 0 是一个二次方程，其中 a = 2，b = −5，c = 3。如果 x² 的系数为零，该方程就不是二次方程。
2. Three Main Solution Methods | 三种主要解法
IGCSE candidates need to be confident with three algebraic methods: factorising, completing the square, and using the quadratic formula. You may also be asked to estimate roots from a graph.
IGCSE 考生需要熟练掌握三种代数方法：因式分解、配方法和使用二次公式。考试也可能要求根据图像估计根。
Factorising is usually quickest when the roots are rational. Completing the square is useful for finding the vertex and solving by rearrangement. The quadratic formula works for every quadratic equation, even when factorising is difficult.
当根是有理数时，因式分解通常最快。配方法适用于求顶点和通过变形求解。二次公式适用于所有二次方程，即使因式分解很困难。
3. Solving by Factorising | 因式分解法
d whose sum is b. Write x² + bx + c = (x + p)(x + q), then set each bracket equal to zero.
要解 x² + bx + c = 0，先找到两个数，它们的乘积是 c，和是 b。写成 x² + bx + c = (x + p)(x + q)，然后令每个括号等于零。
rs −2 and −3 multiply to 6 and add to −5, so (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.
例子：x² − 5x + 6 = 0。−2 和 −3 的乘积是 6，和是 −5，所以 (x − 2)(x − 3) = 0。因此 x = 2 或 x = 3。
When a ≠ 1, use the ac method: find two numbers whose product is ac and sum is b, split the middle term, and factor by grouping.
当 a ≠ 1 时，使用 ac 法：找到两个数，其乘积为 ac，和为 b，拆分中间项，然后分组分解。
4. Completing the Square | 配方法
Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. For x² + bx, add and subtract (b/2)² to create a perfect square.
配方法将 ax² + bx + c 写成 a(x + p)² + q 的形式。对于 x² + bx，加上并减去 (b/2)² 来构造完全平方。
ation becomes (x + 3)² − 4 = 0. Then (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.
例子：x² + 6x + 5 = 0。将 x² + 6x 写成 (x + 3)² − 9，方程变为 (x + 3)² − 4 = 0。于是 (x + 3)² = 4，所以 x + 3 = ±2，得到 x = −1 或 x = −5。
This method also reveals the vertex (−p, q) when the expression is written as a(x + p)² + q.
当表达式写成 a(x + p)² + q 时，该方法还能直接给出顶点 (−p, q)。
5. The Quadratic Formula | 二次公式
For any quadratic ax² + bx + c = 0, the roots are given by:
对于任意二次方程 ax² + bx + c = 0，根由以下公式给出：
x = ( −b ± √b² − 4ac ) / (2a)
This formula is derived from completing the square. It is especially helpful when coefficients are large, negative, or not easy to factorise.
该公式由配方法推导而来。当系数较大、为负数或不容易因式分解时，它特别有用。
, b = 3, c = −2, so b² − 4ac = 9 + 16 = 25. Then x = (−3 ± 5)/4, giving x = 1/2 or x = −2.
例子：2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2，所以 b² − 4ac = 9 + 16 = 25。于是 x = (−3 ± 5)/4，得到 x = 1/2 或 x = −2。
6. Using the Discriminant | 判别式的运用
e discriminant. It tells you how many real roots a quadratic equation has without solving it.
表达式 D = b² − 4ac 称为判别式。它可以在不解方程的情况下判断二次方程有多少个实数根。
real roots. If D = 0, there is one repeated real root. If D < 0, there are no real roots.
如果 D > 0，则有两个不同的实数根。如果 D = 0，则有一个重复的实数根。如果 D < 0，则没有实数根。
Example: x² + 2x + 5 = 0 has D = 4 − 20 = −16, so it has no real roots.
例子：x² + 2x + 5 = 0 的 D = 4 − 20 = −16，因此没有实数根。
7. Graphical Interpretation | 图像理解
arabola. The roots of the equation ax² + bx + c = 0 are the x-intercepts of this parabola.
y = ax² + bx + c 的图像是抛物线。方程 ax² + bx + c = 0 的根是该抛物线与 x 轴的交点横坐标。
arabola touches the x-axis at one point. If D < 0, 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 D = 0，抛物线与 x 轴相切于一点。如果 D < 0，抛物线与 x 轴不相交。
You may be asked to read approximate roots from a graph or to sketch a quadratic given its roots.
考试可能要求你从图像中读取近似根，或根据给定的根画出二次函数草图。
8. Word Problems and Modelling | 应用题与建模
Quadratic equations often model area, projectile motion, and number puzzles. Read the question carefully, define a variable, and form an equation before solving.
二次方程常用于面积、抛体运动和数字谜题的建模。仔细读题，设定变量，先列出方程再求解。
40 = 0. Factorising gives (x + 8)(x − 5) = 0, so x = 5 m because width cannot be negative.
例子：一个矩形的长比宽多 3 米，面积为 40 平方米。设宽为 x，则长为 x + 3。于是 x(x + 3) = 40，得到 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = 5 米，因为宽度不能为负。
Always check that your answer makes sense in the original context, not just in the algebra.
始终检查你的答案在原情境中是否合理，而不仅仅在代数中成立。
9. Common Mistakes to Avoid | 常见错误
For example, do not factorise x² + 3x = 4 as x(x + 3) = 4 and then set factors equal to 4.
在因式分解前忘记将方程设为等于零是一个常见错误。例如，不要把 x² + 3x = 4 因式分解为 x(x + 3) = 4 然后令因式等于 4。
Another mistake is dividing by x when both sides contain x, which can lose a root. Always rearrange and factorise instead.
另一个错误是当两边都含有 x 时除以 x，这可能会丢根。应重新整理并因式分解。
Also, be careful with signs in the quadratic formula, especially when b or c is negative.
此外，使用二次公式时要小心符号，特别是当 b 或 c 为负数时。
10. Exam-style Worked Example | 考试型例题
uestion: Solve 3x² − 7x + 2 = 0 using factorising, and verify using the quadratic formula.
题目：用因式分解法解 3x² − 7x + 2 = 0，并用二次公式验证。
: 3x² − x − 6x + 2 = x(3x − 1) − 2(3x − 1) = (3x − 1)(x − 2) = 0. Roots: x = 1/3 or x = 2.
因式分解：ac = 3 × 2 = 6。需要两个数乘积为 6 且和为 −7：−1 和 −6。拆分：3x² − x − 6x + 2 = x(3x − 1) − 2(3x − 1) = (3x − 1)(x − 2) = 0。根：x = 1/3
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