📚 Mastering Quadratics for Edexcel IGCSE | 掌握二次函数：Edexcel IGCSE数学复习指南
Quadratics are one of the most heavily tested topics in the Edexcel IGCSE Mathematics exam. From factorising to the quadratic formula, from completing the square to sketching graphs, a solid grasp of quadratics can earn you a significant number of marks across both Paper 1 and Paper 2.
二次函数是 Edexcel IGCSE 数学考试中考查频率最高的主题之一。从因式分解到求根公式，从配方法到画图，扎实掌握二次函数能帮助你在 Paper 1 和 Paper 2 中拿到大量分数。
1. The General Form of a Quadratic | 二次函数的一般形式
A quadratic expression is written in the form ax² + bx + c, where a, b, and c are constants and a ≠ 0. The highest power of x is 2, which gives the graph its characteristic U-shape.
二次表达式写作 ax² + bx + c，其中 a、b、c 为常数，且 a ≠ 0。x 的最高次数是 2，这使图像呈现特有的 U 形。
When a > 0, the parabola opens upwards. When a < 0, it opens downwards. The value of c is the y-intercept.
当 a > 0 时，抛物线开口向上；当 a < 0 时，开口向下。c 的值就是 y 轴截距。
- Standard form: ax² + bx + c
- 标准形式：ax² + bx + c
- Roots: values of x where the graph crosses the x-axis
- 根：图像与 x 轴交点的 x 值
2. Expanding and Simplifying | 展开与化简
15. Remember: multiply each term in the first bracket by each term in the second bracket.
展开意味着去掉括号。例如，(x + 3)(x + 5) = x² + 8x + 15。记住：用第一个括号中的每一项去乘以第二个括号中的每一项。
For three terms such as (x + 1)(x + 2)(x + 3), first multiply two factors, then multiply the result by the third.
对于三项如 (x + 1)(x + 2)(x + 3)，先乘两个因式，再将结果乘第三个因式。
(x + 3)² = x² + 6x + 9
You should also be comfortable expanding expressions like (2x − 1)(x + 4) = 2x² + 7x − 4.
你还应熟练展开如 (2x − 1)(x + 4) = 2x² + 7x − 4 这样的表达式。
3. Factorising Quadratics | 二次因式分解
Factorising is the reverse of expanding. To factorise x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆运算。要分解 x² + bx + c，需要找到两个数，它们相乘等于 c，相加等于 b。
e −2 and −3 because (−2) × (−3) = 6 and (−2) + (−3) = −5. So x² − 5x + 6 = (x − 2)(x − 3).
例如，分解 x² − 5x + 6。这两个数是 −2 和 −3，因为 (−2) × (−3) = 6 且 (−2) + (−3) = −5。所以 x² − 5x + 6 = (x − 2)(x − 3)。
When a ≠ 1, use the method of factors of a and c, or the 'cross method'. For 2x² + 7x + 3, write (2x + 1)(x + 3).
当 a ≠ 1 时，可使用系数 a 和 c 的因数法，或交叉相乘方法。对于 2x² + 7x + 3，可写成 (2x + 1)(x + 3)。
Always check by expanding your answer.
务必通过展开来检查你的答案。
4. Solving Quadratic Equations by Factorising | 利用因式分解解二次方程
equals zero, you can solve it by factorising. If (x − p)(x − q) = 0, then x = p or x = q.
如果二次方程等于零，你可以通过因式分解来求解。若 (x − p)(x − q) = 0，则 x = p 或 x = q。
Example: solve x² − 3x − 10 = 0. Factorise: (x − 5)(x + 2) = 0. Therefore x = 5 or x = −2.
示例：解 x² − 3x − 10 = 0。因式分解得 (x − 5)(x + 2) = 0。因此 x = 5 或 x = −2。
not in the form = 0, rearrange it first. For example, x² = 4x + 5 becomes x² − 4x − 5 = 0.
如果方程不是等于 0 的形式，先移项。例如，x² = 4x + 5 变为 x² − 4x − 5 = 0。
5. The Quadratic Formula | 求根公式
n factorising is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0:
当因式分解困难或无法进行时，使用求根公式。对于 ax² + bx + c = 0：
x = (−b ± √b² − 4ac) / 2a
You must memorise this formula. It is not given on the Edexcel IGCSE formula sheet.
你必须记住这个公式。Edexcel IGCSE 的公式表中不会给出它。
Example: solve 2x² + 5x − 3 = 0. Here a = 2, b = 5, c = −3. Substitute:
示例：解 2x² + 5x − 3 = 0。这里 a = 2，b = 5，c = −3。代入：
x = (−5 ± √(25 − 4×2×(−3))) / 4 = (−5 ± √49) / 4 = (−5 ± 7) / 4
So x = 2/4 = 0.5 or x = −12/4 = −3.
因此 x = 2/4 = 0.5 或 x = −12/4 = −3。
6. The Discriminant | 判别式
the square root: Δ = b² − 4ac. It tells you how many real roots a quadratic equation has.
判别式是根号内的部分：Δ = b² − 4ac。它告诉我们二次方程有多少个实数根。
| Δ = b² − 4ac | Number of real roots | 实数根个数 |
| Δ > 0 | Two distinct real roots | 两个不同的实数根 |
| Δ = 0 | One repeated real root | 一个重根 |
| Δ < 0 | No real roots | 没有实数根 |
If the question asks for 'roots', it usually means real roots. You can also use the discriminant to determine whether a quadratic intersects, touches, or misses the x-axis.
如果题目问“根”，通常指实数根。你也可以用判别式判断二次函数的图像与 x 轴相交、相切还是相离。
7. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful for finding the vertex (turning point) and for solving equations.
配方法将二次式改写为 a(x + p)² + q 的形式。它有助于找到顶点（转折点）和解方程。
For x² + bx + c, take half of b, square it, and adjust:
对于 x² + bx + c，取 b 的一半，平方，然后调整：
x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7
If a ≠ 1, first factor a out of the x² and x terms before completing the square.
如果 a ≠ 1，先将 a 从 x² 和 x 项中提出来，再进行配方。
8. Solving by Completing the Square | 用配方法解方程
ed square form a(x + p)² + q = 0, you can solve it by rearranging and taking square roots.
一旦二次方程成为 a(x + p)² + q = 0 的形式，你可以通过移项并开平方来求解。
Example: solve (x + 3)² − 7 = 0.
示例：解 (x + 3)² − 7 = 0。
(x + 3)² = 7 → x + 3 = ±√7 → x = −3 ± √7
This gives two irrational roots. In IGCSE exams, you may leave your answer in surd form.
这给出两个无理根。在 IGCSE 考试中，答案可以保留根号形式。
9. Sketching Quadratic Graphs | 二次函数图像的草图
To sketch a quadratic graph, you need three key features: the y-intercept, the roots (if any), and the turning point.
要画二次函数图像的草图，你需要三个关键特征：y 轴截距、根（如果有）以及转折点。
- y-intercept: set x = 0, find y = c
- y 轴截距：令 x = 0，得到 y = c
- Roots: set y = 0, solve for x
- 根：令 y = 0，解 x
- Turning point: use completed square form (p, q) or x = −b/(2a)
- 转折点：使用配方形式 (p, q) 或 x = −b/(2a)
3. y-intercept at (0, 3). Complete the square: (x − 2)² − 1, so the minimum point is (2, −1).
示例：画 y = x² − 4x + 3 的草图。因式分解得 (x − 1)(x − 3)，所以根在 x = 1 和 x = 3。y 轴截距为 (0, 3)。配方得 (x − 2)² − 1，所以最低点为 (2, −1)。
10. Applications: Word Problems | 应用：文字题
Quadratic equations often appear in geometric or real-life problems. For example, the area of a rectangle is given by a quadratic expression.
二次方程经常出现在几何或实际生活问题中。例如，矩形的面积由二次表达式给出。
Problem: A rectangle's length is 3 cm longer than its width, and its area is 28 cm². Find its width.
问题：一个矩形的长比宽长 3 cm，面积为 28 cm²。求它的宽。
xpand: x² + 3x − 28 = 0. Factorise: (x + 7)(x − 4) = 0. Since width is positive, x = 4 cm.
设宽为 x，则长为 x + 3，且 x(x + 3) = 28。展开得 x² + 3x − 28 = 0。因式分解得 (x + 7)(x − 4) = 0。由于宽为正数，x = 4 cm。
Always reject negative solutions for lengths, times, or counts.
对于长度、时间或数量，务必舍去负数解。
11. Common Exam Traps | 常见考试陷阱
Many marks are lost through careless errors. Watch out for these common mistakes:
很多分数因粗心错误而丢失。请注意以下常见错误：
- Forgetting that a ≠ 0 in a quadratic
- 忘记二次函数中 a ≠ 0
- Mixing up signs when factorising (x − 2)(x − 3)
- 因式分解时弄错符号，如 (x − 2)(x − 3)
- Using the quadratic formula without writing the equation in the form ax² + bx + c = 0
- 没有先将方程写成 ax² + bx + c = 0 就使用求根公式
- Forgetting to include ± when taking square roots
- 开平方时忘记写 ±
- Rounding prematurely when the answer should be in surd form
- 当答案应为根号形式时过早四舍五入
12. Practice Quick Questions | 快速练习
Try these questions to test yourself. Answers are shown below.
试试这些题目来测试自己。答案在下方。
- Factorise x² + 7x + 12
- Solve 3x² − 5x − 2 = 0
- Write y = x² + 8x + 5 in completed square form
- Find the discriminant of 2x² + 3x + 1 and state the number of real roots
- Sketch the graph of y = x² − 2x − 3, labelling the roots, y-intercept, and turning point
real roots 5. Roots at x = 3 and x = −1; y-intercept at (0, −3); turning point at (1, −4)
答案：1. (x + 3)(x + 4) 2. x = 2 或 x = −1/3 3. (x + 4)² − 11 4. Δ = 1，两个实数根 5. 根为 x = 3 和 x = −1；y 截距为 (0, −3)；转折点为 (1, −4)
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