📚 Mastering Quadratic Equations and Graphs | 掌握二次方程与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebra and graphs to real-life modelling problems. This revision guide walks you through the standard form, solution methods, graph features and typical exam traps. Understanding quadratics deeply will also prepare you for further topics such as inequalities and transformations.
二次方程贯穿 IGCSE 数学大纲，从代数、函数图像到实际问题建模都会涉及。本复习指南带你系统梳理标准形式、求解方法、图像特征以及常见考试陷阱。扎实掌握二次方程还能为你后续学习不等式、图像变换等内容打下基础。
1. What is a Quadratic Equation? | 什么是二次方程？
onstants and a ≠ 0. If a = 0, the equation becomes linear and must be treated differently.
二次方程是次数为 2 的多项式方程。它总可以写成 ax² + bx + c = 0 的形式，其中 a、b、c 为常数且 a ≠ 0。如果 a = 0，方程就退化为一次方程，需要按一次方程处理。
actly the x-intercepts of the graph, which is why solving and graphing are closely linked.
y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时开口向上，当 a < 0 时开口向下。方程 ax² + bx + c = 0 的解就是图像与 x 轴交点的横坐标，因此求解与画图密切相关。
7 = 0 is actually the linear equation 5x − 7 = 0, so a must never be zero in a quadratic.
例如：3x² − 2x + 1 = 0 是二次方程，因为 x 的最高次数为 2。而 0x² + 5x − 7 = 0 其实是一次方程 5x − 7 = 0，所以二次方程中 a 绝不能为零。
2. Standard Form and Key Terms | 标准形式与关键术语
g coefficient, b is the linear coefficient, and c is the constant term. Collect like terms and set one side to zero first.
求解之前，务必先把二次方程整理成标准形式：ax² + bx + c = 0。系数 a 是二次项系数，b 是一次项系数，c 是常数项。先合并同类项，并让等式一边等于零。
Key vocabulary includes roots, solutions, zeros and x-intercepts. These all refer to the values of x that make the quadratic expression equal to zero. In IGCSE questions, you may see any of these terms, so know them well.
关键术语包括根、解、零点和 x 轴截距。它们都指使二次代数式等于零的 x 值。在 IGCSE 考题中，这些术语都可能出现，因此要熟练掌握。
| Equation | a | b | c |
|---|---|---|---|
| 2x² + 5x − 3 = 0 | 2 | 5 | −3 |
| x² − 4x + 7 = 0 | 1 | −4 | 7 |
| −3x² + 2 = 0 | −3 | 0 | 2 |
The table above shows how to identify a, b and c from different equations. Notice that a can be positive or negative, b can be zero, and c can be zero as well, but a cannot be zero.
上表展示了如何从不同方程中识别 a、b 和 c。注意 a 可以为正或负，b 可以为零，c 也可以为零，但 a 不能为零。
3. Solving by Factorising | 因式分解法
factor to zero. For example, x² − 5x + 6 = 0 gives (x − 2)(x − 3) = 0, so x = 2 or x = 3.
当一个二次方程可以写成两个一次因式的乘积时，我们可以令每个因式等于零。例如 x² − 5x + 6 = 0 可化为 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。
he roots are integers or simple fractions. For harder coefficients, use the ac method or move to another method. Always check that you have set the equation to zero before factorising.
当 a = 1 且根为整数或简单分数时，因式分解法最有效。对于较复杂的系数，可以利用 ac 方法，或改用其他方法。因式分解前一定要确认方程已经化为等于零的形式。
2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0. Therefore x = −½ or x = −3.
例如：解 2x² + 7x + 3 = 0。先求 ac = 2 × 3 = 6。因数 6 和 1 相加为 7，所以把 7x 拆成 6x + x。然后分组：2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3) = 0。因此 x = −½ 或 x = −3。
time forcing a factorisation. Use the quadratic formula or completing the square instead.
如果二次方程不容易因式分解，例如 x² − 3x − 5 = 0，不要浪费时间强行分解。此时应改用求根公式或配方法。
4. Solving by Completing the Square | 配方法
Completing the square transforms x² + bx into (x + b/2)² − (b/2)². This is especially useful when the quadratic does not factorise nicely, and it reveals the turning point of the graph directly.
配方法将 x² + bx 变形为 (x + b/2)² − (b/2)²。当二次方程不方便因式分解时，配方法尤其有用，而且它能直接揭示图像的顶点位置。
)² − 9 + 5 = 0, so (x + 3)² = 4. Take the square root: x + 3 = ±2. Hence x = −1 or x = −5.
例如：x² + 6x + 5 = 0。将 x² + 6x 写成 (x + 3)² − 9。方程就变为 (x + 3)² − 9 + 5 = 0，即 (x + 3)² = 4。开平方：x + 3 = ±2。所以 x = −1 或 x = −5。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图