📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：二次方程求解
Quadratic equations are a central topic in IGCSE Mathematics. They appear in algebra, coordinate geometry and real-life modelling, so a confident method is essential for a high grade.
二次方程是 IGCSE 数学的核心主题。它们出现在代数、坐标几何和实际建模中，因此掌握可靠的解法对取得高分至关重要。
1. Understanding Quadratic Equations | 理解二次方程
+ c = 0, where a, b and c are constants and a ≠ 0. The x² term is what makes the equation 'quadratic'.
二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。x² 项使得该方程成为“二次”。
0, 2x² + 3x − 2 = 0 and x² − 9 = 0. If a = 0, the equation becomes linear, not quadratic.
示例包括 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 x² − 9 = 0。如果 a = 0，方程就变成一次方程，而不是二次方程。
2. Standard Form and Key Terms | 标准形式与关键术语
ore solving. The values of a, b and c are used in factorisation and the quadratic formula.
求解前始终将二次方程整理为标准形式 ax² + bx + c = 0。a、b、c 的值将用于因式分解和求根公式。
Key terms include the roots, which are the x-values that make the equation true, and the discriminant b² − 4ac, which tells us how many real roots exist.
关键术语包括根，即令方程成立的 x 值；以及判别式 b² − 4ac，它告诉我们存在多少个实根。
The leading coefficient a must not be zero, and the equation is usually solved over the real numbers unless the question states otherwise.
首项系数 a 不能为零，并且除非题目另有说明，方程通常是在实数范围内求解。
3. Solving by Factorisation | 因式分解法
xpressed as (px + q)(rx + s) = 0. You then set each bracket equal to zero and solve for x.
当二次方程可以表示为 (px + q)(rx + s) = 0 时，可以使用因式分解法。然后令每个括号等于零并求解 x。
ply to +6 and add to −5: they are −2 and −3. So (x − 2)(x − 3) = 0, giving x = 2 or x = 3.
对于 x² − 5x + 6 = 0，找到两个相乘为 +6、相加为 −5 的数：它们是 −2 和 −3。因此 (x − 2)(x − 3) = 0，得到 x = 2 或 x = 3。
If a ≠ 1, use the 'ac method' − 2, look for factors of 2 and −2 that give the correct middle term: (2x − 1)(x + 2) = 0.
如果 a ≠ 1，可使用 "ac 法" 或尝试法。对于 2x² + 3x − 2，寻找 2 和 −2 的因数以获得正确的中间项：(2x − 1)(x + 2) = 0。
Always expand your brackets as a check. If the expansion returns to the original quadratic, the factorisation is likely correct.
始终通过展开括号来检验。如果展开后回到原二次方程，则因式分解很可能是正确的。
4. Solving by 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 factorisation is difficult.
配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。当因式分解困难时，该方法尤其有用。
6 to get 3, then write (x + 3)² − 9 + 2 = 0. This simplifies to (x + 3)² = 7, so x = −3 ± √7.
对于 x² + 6x + 2 = 0，取 6 的一半得到 3，然后写成 (x + 3)² − 9 + 2 = 0。这化简为 (x + 3)² = 7，因此 x = −3 ± √7。
When a ≠ 1, factor out a first. This method also helps to find the vertex of a parabola: the turning point is (−p, q).
当 a ≠ 1 时，先提取 a。此方法也有助于求抛物线的顶点：转折点为 (−p, q)。
In IGCSE questions, completing the square is often tested on its own, so make sure you can write the completed-square form accurately.
在 IGCSE 题目中，配方法经常单独考查，因此务必确保能够准确写出完全平方形式。
5. Solving by the Quadratic Formula | 求根公式法
hat do not factorise neatly. It is derived from completing the square on ax² + bx + c = 0.
求根公式适用于所有二次方程，包括那些不能整齐因式分解的方程。它由对 ax² + bx + c = 0 配方推导而来。
x = (−b ± √b² − 4ac) / 2a
For 2x² + 3x − 2 = 0, a = 2, b = 3, c = −2. Substituting gives x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4, so x = ½ or x = −2.
对于 2x² + 3x − 2 = 0，a = 2，b = 3，c = −2。代入得到 x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4，因此 x = ½ 或 x = −2。
Memorise the formula and practise using it with negative values of b and c. Always write the formula first in an exam to avoid sign errors.
记住该公式，并练习在 b 和 c 为负值的情况下使用它。考试时务必先写下公式，以避免符号错误。
When a calculator is allowed, store intermediate values in memory or use the fraction button to keep full precision until the final answer.
在允许使用计算器时，应利用存储功能或分数键保持中间值的完整精度，直到最终答案。
6. Discriminant and Nature of Roots | 判别式与根的性质
ls us whether a quadratic has two distinct real roots, one repeated real root, or no real roots.
判别式为 D = b² − 4ac。它告诉我们二次方程有两个不同实根、一个重根还是没有实根。
- D > 0: two distinct real roots — D > 0：两个不同实根
- D = 0: one repeated real root — D = 0：一个重根
- D < 0: no real roots — D < 0：无实根
IGCSE questions often ask you to determine the number of roots without solving, so always check the discriminant when needed.
IGCSE 题目经常要求你在不求解的情况下确定根的数量，因此需要时务必检查判别式。
If D is a perfect square, the quadratic has rational roots, which means it can be factorised using integers or simple fractions.
如果 D 是一个完全平方数，则该二次方程有有理根，这意味着它可以用整数或简单分数进行因式分解。
7. Applications and Word Problems | 实际应用与文字题
Quadratic equations model many real-life situations such as area, projectile motion and profit. You usually define a variable, set up an equation, then solve it and check the context.
二次方程可模拟许多实际情境，如面积、抛体运动和利润。通常需要定义变量、建立方程，然后求解并根据情境检验。
(x + 3) = 40, giving x² + 3x − 40 = 0. Solving gives x = 5 or x = −8; only x = 5 is valid.
例如，若矩形宽为 x m，长为 (x + 3) m，面积为 40 m²，则 x(x + 3) = 40，得到 x² + 3x − 40 = 0。解得 x = 5 或 x = −8；只有 x = 5 有效。
Always reject negative lengths, times or areas in real-world problems. State the answer with correct units.
在实际问题中，务必舍去负的长度、时间或面积。答案应附上正确单位。
= ut + ½at². Substitute the known values first, then simplify to standard quadratic form.
有时二次方程来自 s = ut + ½at² 等公式。应先代入已知数值，然后化简为标准二次方程形式。
8. Common Mistakes and Exam Tips | 常见错误与考试技巧
A common mistake is forgetting to set the equation to zero before factor
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