📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
This revision guide covers the essential IGCSE topic of quadratic equations and their graphs. You will learn how to solve quadratics by factorisation, completing the square, and the quadratic formula, and how to interpret the graph of y = ax² + bx + c. These skills 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？
A quadratic equation in one variable is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic. The highest 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
Factorisation is often the fastest method when the quadratic factorises into two simple brackets. Write the equation in the form ax² + bx + c = 0, then 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
Find two numbers whose product is 6 and sum is 5. The numbers are 2 and 3, so we can write (x + 2)(x + 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
Move the constant: x² + 6x = -5. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 4. This gives (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
The quadratic formula solves every quadratic equation, 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
Here a = 2, b = -4, c = -3. Substituting into 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
The expression D = b² – 4ac is called 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
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point. The y-intercept 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
The axis of symmetry of the parabola is the vertical line x = -b / (2a). The vertex lies on this axis. To find the vertex, substitute x = -b / (2a) into the equation. Alternatively, complete the square to write y = a(x – h)² + k; then the vertex is (h, k).
抛物
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