📚 Mastering Quadratic Equations for IGCSE Maths | 掌握 IGCSE 数学二次方程
A quadratic equation appears in many IGCSE Maths topics, from algebra to area problems and projectile motion. This revision guide covers the key methods you need: rearranging, factorising, using the quadratic formula, understanding the discriminant, completing the square, and avoiding common errors.
二次方程出现在 IGCSE 数学的许多主题中，从代数到面积问题和抛体运动。本复习指南涵盖你需要掌握的关键方法：移项、因式分解、使用二次公式、理解判别式、配方法以及避免常见错误。
1. What Is a Quadratic Equation? | 什么是二次方程？
0, where a, b, and c are constants and a ≠ 0. The term quadratic comes from the Latin word 'quadratus', meaning square, because the highest power of the unknown x is 2.
二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。quadratic 一词源于拉丁语 'quadratus'，意为“平方”，因为未知数 x 的最高次数是 2。
2. Standard Form and Coefficients | 标准形式与系数
a = 3, b = -2, and c = -5. Writing the standard form helps you choose the correct method.
求解前始终要把方程整理为右边等于 0 的形式。例如 3x² - 5 = 2x 化为 3x² - 2x - 5 = 0，得到 a = 3、b = -2、c = -5。写成标准形式有助于你选择正确的解法。
3. Solving by Factorising | 因式分解法
to 0. This works because of the zero product property: if A × B = 0, then A = 0 or B = 0.
如果二次三项式可因式分解为 (px + q)(rx + s) = 0，那么令每个括号分别等于 0 即可求解。这是因为零积性质：若 A × B = 0，则 A = 0 或 B = 0。
4. Worked Example: Factorising | 例题：因式分解
12 = (x + 3)(x + 4). Set each factor to 0: x + 3 = 0 gives x = -3; x + 4 = 0 gives x = -4.
解 x² + 7x + 12 = 0。找两个数，乘积为 +12、和为 +7。数对 (1,12)、(2,6)、(3,4) 中 3 和 4 符合，因此 x² + 7x + 12 = (x + 3)(x + 4)。令每个因式为 0：x + 3 = 0 得 x = -3；x + 4 = 0 得 x = -4。
5. Quadratic Formula | 二次公式法
torising is difficult, use the quadratic formula. For ax² + bx + c = 0, the solutions are:
当因式分解较困难时使用二次公式。对于 ax² + bx + c = 0，解为：
x = (-b ± √b² - 4ac) / (2a)
ire denominator 2a. Many marks are lost by writing only one sign or by dividing partially.
准确记住这个公式，包括 ± 号和整个分母 2a。许多考生因为只写一个符号或只部分除以分母而丢分。
6. Worked Example: Quadratic Formula | 例题：二次公式
= 0. Here a = 2, b = 3, c = -5. Substituting: x = (-3 ± √(3² - 4×2×(-5))) / (2×2) = (-3 ± √9 + 40) / 4 = (-3 ± √49) / 4 = (-3 ± 7) / 4. So x = 1 or x = -2.5.
解 2x² + 3x - 5 = 0。此处 a = 2、b = 3、c = -5。代入：x = (-3 ± √(3² - 4×2×(-5))) / (2×2) = (-3 ± √9 + 40) / 4 = (-3 ± √49) / 4 = (-3 ± 7) / 4。所以 x = 1 或 x = -2.5。
7. Discriminant and Roots | 判别式与根
is D = b² - 4ac. It tells you the number of real roots without solving the equation fully.
判别式为 D = b² - 4ac。它可以在完全解方程之前告诉你实根的个数。
| D > 0 | two distinct real roots | 两个不同实根 |
| D = 0 | one repeated real root | 一个重根 |
| D < 0 | no real roots | 没有实根 |
IGCSE questions often ask you to state how many roots an equation has, so always calculate D first if root counting is required.
IGCSE 题目经常要求说明方程有多少个根，因此如果只需求根的数量，始终先计算 D。
8. Completing the Square | 配方法
² + bx + c, add and subtract (b/2)². The vertex of the parabola is (h, k), which helps with graph sketches.
配方法将二次式改写为 a(x - h)² + k = 0 的形式。对 x² + bx + c，加减 (b/2)² 即可。抛物线的顶点为 (h, k)，这有助于画图。
square. x² + 6x + 9 - 9 + 2 = 0 becomes (x + 3)² - 7 = 0, so (x + 3)² = 7, giving x = -3 ± √7.
用配方法解 x² + 6x + 2 = 0。x² + 6x + 9 - 9 + 2 = 0 化为 (x + 3)² - 7 = 0，因此 (x + 3)² = 7，解得 x = -3 ± √7。
9. Word Problems | 应用题策略
Translate the problem into a quadratic equation. Identify the unknown, write expressions for quantities, and use the given condition. Check that answers make sense in context; negative lengths or times may need rejecting.
将实际问题转化为二次方程。设未知数、表示相关量、利用条件列方程。检查答案在实际情境中是否合理，长度或时间的负值通常需要舍去。
ength is 2 cm more than its width, let width = x, length = x + 2, and solve x(x + 2) = 24.
例如，某长方形面积为 24 cm²，长比宽多 2 cm。设宽为 x，长为 x + 2，解方程 x(x + 2) = 24。
10. Common Mistakes | 常见错误
ly. For example, do not divide both sides of x² = 5x by x because x = 0 may be a solution.
常见错误包括因式分解前忘记让方程右边为 0、公式中漏掉 ±、错误地约去变量。例如不要将 x² = 5x 两边同除以 x，因为 x = 0 也可能是解。
Another mistake is writing the denominator as a instead of 2a. Always substitute values in brackets to keep negative signs clear.
另一个错误是把分母写成 a 而不是 2a。代入数值时要加括号，避免负号出错。
11. Exam Tips | 考试技巧
In the IGCSE exam, show all steps, check your factorisation by expanding, and use the formula if factorising takes more than one minute. Write approximate answers to 3 significant figures when asked.
在 IGCSE 考试中，写出完整步骤、通过展开检验因式分解、若因式分解超过一分钟就改用公式。需要近似值时按题目要求保留三位有效数字。
When a question asks for exact answers, leave surds such as -3 ± √7, not rounded decimals. If it asks for solutions to 3 s.f., then round at the end of the calculation.
如果题目要求精确答案，保留根号，如 -3 ± √7，不要写成四舍五入的小数。如果要求保留三位有效数字，则在计算最后才取近似值。
12. Summary | 总结
Master rearranging, factorising, the quadratic formula, the discriminant, and completing the square. Practise mixed exercises to build speed and accuracy. Recognising which method is fastest comes from repeated timed practice.
掌握移项、因式分解、二次公式、判别式与配方法。通过混合练习提高速度与准确性。通过限时重复训练，你才能快速判断哪种方法最快。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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