📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
lls appear frequently in both core and extended papers and are often linked to real-life problems.
本复习指南涵盖 IGCSE 二次方程及其图像这一核心主题。你将学习如何用因式分解、配方法和求根公式解二次方程，以及如何理解 y = ax² + bx + c 的图像。这些技能在核心卷和扩展卷中都经常出现，并且常常与实际问题相结合。
1. What is a Quadratic Equation? | 什么是二次方程？
est power of x is 2. For example, 2x² - 5x + 3 = 0 is quadratic, but 3x - 7 = 0 is linear.
一元二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。若 a = 0，方程就变成一次方程，而不是二次方程。x 的最高次数是 2。例如 2x² - 5x + 3 = 0 是二次方程，但 3x - 7 = 0 是一次方程。
A quadratic equation has at most two real solutions, because the graph of a quadratic function can cross the x-axis at most twice. Some equations have one repeated solution, and some have no real solutions at all.
一元二次方程最多有两个实数解，因为二次函数的图像最多与 x 轴相交两次。有些方程有一个重根，有些方程则没有实数解。
2. Solving by Factorisation | 因式分解法
find two numbers that multiply to give ac and add to give b. Split the middle term, factor by grouping, and set each factor equal to zero.
当二次式能分解成两个简单括号时，因式分解通常是最快的方法。先把方程写成 ax² + bx + c = 0，然后找到两个数，它们相乘得 ac，相加得 b。拆中项，分组分解，再令每个因式等于零。
Solve: x² + 5x + 6 = 0
3) = 0. Setting each bracket to zero gives x + 2 = 0 or x + 3 = 0, hence x = -2 or x = -3.
找两个数，乘积为 6，和为 5。这两个数是 2 和 3，所以可写成 (x + 2)(x + 3) = 0。令每个括号等于零，得到 x + 2 = 0 或 x + 3 = 0，因此 x = -2 或 x = -3。
Always check that the equation is equal to zero before factorising. If it is not, rearrange all terms to one side first.
在因式分解之前，务必检查方程是否等于零。如果不等于零，要先把所有项移到一边。
3. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x - h)² + k. This form shows the vertex of the parabola and is useful when a quadratic does not factorise easily. For x² + bx, add and subtract (b/2)² to create a perfect square.
配方法把二次式改写为 a(x - h)² + k 的形式。这种形式能直接显示抛物线的顶点，在二次式不容易因式分解时非常有用。对于 x² + bx，加上并减去 (b/2)²，形成完全平方。
Solve: x² + 6x + 5 = 0
(x + 3)² = 4. Taking the square root of both sides gives x + 3 = ±2, so x = -1 or x = -5.
把常数项移到右边：x² + 6x = -5。两边加上 (6/2)² = 9：x² + 6x + 9 = 4。于是有 (x + 3)² = 4。两边开平方得 x + 3 = ±2，所以 x = -1 或 x = -5。
This method also helps when the coefficient of x² is not 1. First divide every term by a, then complete the square as usual.
当 x² 的系数不是 1 时，此方法同样适用。先用 a 除以每一项，再按照通常的方法配方。
4. The Quadratic Formula | 求根公式
ation, even those that do not factorise. For ax² + bx + c = 0, the solutions are given by:
求根公式能解所有二次方程，包括无法因式分解的方程。对于 ax² + bx + c = 0，解为：
x = (-b ± √b² - 4ac) / (2a)
Remember to substitute the values of a, b and c carefully, including negative signs. A common error is to forget that -b means the sign of b changes.
代入 a、b、c 时要格外小心，尤其是负号。常见错误是忘记 -b 表示 b 的符号要改变。
Solve: 2x² - 4x - 3 = 0
the formula gives x = (4 ± √((-4)² - 4 × 2 × (-3))) / (2 × 2). This simplifies to x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2.
这里 a = 2，b = -4，c = -3。代入公式得 x = (4 ± √((-4)² - 4 × 2 × (-3))) / (2 × 2)。化简为 x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4 = (2 ± √10) / 2。
5. The Discriminant | 判别式
d the discriminant. It tells us the number and type of roots without solving the equation.
表达式 D = b² - 4ac 称为判别式。它能在不解方程的情况下判断根的数量和类型。
- If D > 0, there are two distinct real roots.
- If D = 0, there is one repeated real root.
- If D < 0, there are no real roots.
- 若 D > 0，有两个不相等的实根。
- 若 D = 0，有一个重根。
- 若 D < 0，没有实根。
In IGCSE exams, you may be asked to comment on the discriminant or to find the range of values for which a quadratic has real roots.
在 IGCSE 考试中，你可能会被要求利用判别式判断根的情况，或求二次方程有实根时参数的取值范围。
6. Graphs of Quadratic Functions | 二次函数图像
ercept is (0, c). The x-intercepts, if they exist, are the real roots of ax² + bx + c = 0.
y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上，有最小值点；若 a < 0，开口向下，有最大值点。y 轴截距是 (0, c)。如果存在 x 轴截距，它们就是 ax² + bx + c = 0 的实根。
Sketching a quadratic graph requires labelling the y-intercept, the x-intercepts if they exist, and the vertex. This is a common IGCSE examination skill.
画二次函数草图时，需要标出 y 轴截距、x 轴截距（如果存在）以及顶点。这是 IGCSE 考试的常见技能。
7. Vertex and Axis of Symmetry | 顶点与对称轴
Alternatively, complete the square to write y = a(x - h)² + k; then the vertex is (h, k).
抛物
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