📚 IGCSE Maths: Quadratic Equations and Graphs | IGCSE 数学：二次方程与图像
Quadratic equations appear throughout the IGCSE Mathematics syllabus, from straightforward factorisation to graph sketching and real-life modelling. This revision guide brings together the essential methods, common pitfalls and exam techniques you need to solve quadratic problems confidently.
二次方程贯穿 IGCSE 数学课程，从简单的因式分解到图像绘制和实际建模。本复习指南汇总了核心方法、常见错误和考试技巧，帮助你自信地解答二次方程相关问题。
1. What is a Quadratic Equation? | 什么是二次方程？
stants and a ≠ 0. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.
二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。ax² 是二次项，bx 是一次项，c 是常数项。
so the condition a ≠ 0 is essential. IGCSE questions may ask you to recognise the standard form or to rearrange an equation into this form before solving.
如果 a = 0，方程就变成了一次方程，因此条件 a ≠ 0 至关重要。IGCSE 题目可能要求你识别标准形式，或在求解前先把方程整理成这种形式。
2. Expanding and Factorising Quadratics | 展开与因式分解二次式
Factorising is usually the fastest method for solving a quadratic equation. To factorise x² + bx + c, look for two numbers that multiply to give c and add to give b.
因式分解通常是解二次方程最快的方法。要对 x² + bx + c 进行因式分解，需要找到两个数，它们的乘积等于 c，同时它们的和等于 b。
), because 2 × 5 = 10 and 2 + 5 = 7. Always expand the brackets back to check your answer.
例如，x² + 7x + 10 可以因式分解为 (x + 2)(x + 5)，因为 2 × 5 = 10，且 2 + 5 = 7。务必再展开括号来检查答案。
When the leading coefficient is not 1, such as in 2x² + 5x + 3, split the middle term or use trial and error. The correct factorisation is (2x + 3)(x + 1).
当首项系数不是 1 时，例如 2x² + 5x + 3，可以拆分中间项或使用尝试法。正确的因式分解结果是 (2x + 3)(x + 1)。
3. Solving by Factorisation | 因式分解法解二次方程
if (x + p)(x + q) = 0, then x + p = 0 or x + q = 0. This gives the two solutions directly.
一旦二次式被写成两个括号的乘积，就可以使用零积法则：如果 (x + p)(x + q) = 0，那么 x + p = 0 或 x + q = 0。这样就可以直接得到两个解。
)(x − 3) = 0, so x = 2 or x = 3. Always give exact answers and check them by substitution.
例如，解 x² − 5x + 6 = 0，得到 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。始终给出精确答案，并通过代入原方程进行检验。
Remember that a quadratic equation can have two solutions, one repeated solution, or no real solutions. Factorisation works best when the roots are integers or simple fractions.
请记住，二次方程可以有两个解、一个重根，或者没有实数解。当根是整数或简单分数时，因式分解法效果最好。
4. The Quadratic Formula | 二次公式求根
is difficult, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by:
当因式分解比较困难时，使用二次公式。对于 ax² + bx + c = 0，其解为：
x = (−b ± √b² − 4ac) / (2a)
ions: one using the plus sign and one using the minus sign. Always write the formula from memory before substituting values.
符号 ± 表示有两个解：一个用加号，一个用减号。在代入数值之前，务必先凭记忆写出公式。
2x² + 3x − 2 = 0 gives a = 2, b = 3, c = −2. Substituting into the formula gives x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4, so x = 1/2 or x = −2.
例如，解 2x² + 3x − 2 = 0，令 a = 2，b = 3，c = −2。代入公式得 x = (−3 ± √9 + 16) / 4 = (−3 ± 5) / 4，因此 x = 1/2 或 x = −2。
5. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form a(x + p)² + q. This is especially useful for finding the vertex of a graph or solving equations where factorisation is not obvious.
配方法可以将二次式改写为 a(x + p)² + q 的形式。这在求图像顶点或解不易因式分解的方程时特别有用。
ch is 3, and square it to get 9. Then write x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.
对于 x² + 6x + 5，取 x 项系数的一半 3，并平方得到 9。然后写成 x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。
0, rearrange to (x + 3)² = 4, then take square roots: x + 3 = ±2, giving x = −1 or x = −5.
要解 (x + 3)² − 4 = 0，先整理为 (x + 3)² = 4，再开平方：x + 3 = ±2，得到 x = −1 或 x = −5。
6. The Discriminant | 判别式
The discriminant is the expression b² − 4ac inside the square root of the quadratic formula. It tells you how many real solutions a quadratic equation has without solving it.
判别式是二次公式中平方根内的表达式 b² − 4ac。它可以在不求解方程的情况下判断二次方程有多少个实数解。
² − 4ac = 0, there is one repeated real root. If b² − 4ac < 0, there are no real roots.
如果 b² − 4ac > 0，则有两个不同的实数根。如果 b² − 4ac = 0，则有一个重根。如果 b² − 4ac < 0，则没有实数根。
nant 4² − 4 × 1 × 5 = 16 − 20 = −4, so the equation x² + 4x + 5 = 0 has no real solutions.
例如，x² + 4x + 5 的判别式为 4² − 4 × 1 × 5 = 16 − 20 = −4，因此方程 x² + 4x + 5 = 0 没有实数解。
7. Graphs of Quadratic Functions | 二次函数图像
. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.
y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，并有一个最低点。如果 a < 0，抛物线开口向下，并有一个最高点。
ecause when x = 0, y = c. The x-intercepts are the roots of the equation ax² + bx + c = 0.
y 轴截距是常数项 c，因为当 x = 0 时，y = c。x 轴截距是方程 ax² + bx + c = 0 的根。
Sketching a quadratic graph requires labelling the intercepts, the turning point, and the correct shape. IGCSE exams often award marks for the general shape rather than precise plotting.
画二次函数图像草图时，需要标出截距、顶点和正确的形状。IGCSE 考试通常更看重大致形状，而不是精确的逐点绘制。
8. Turning Point and Axis of Symmetry | 顶点与对称轴
ven by x = −b / (2a). The axis of symmetry is the vertical line through this x-coordinate.
抛物线的顶点是其最高点或最低点。其 x 坐标由 x = −b / (2a) 给出。对称轴是经过这个 x 坐标的竖直线。
ample, y = (x − 3)² − 4 has a turning point at (3, −4), and the axis of symmetry is x = 3.
在配方法形式 y = a(x + p)² + q 中，顶点为 (−p, q)。例如，y = (x − 3)² − 4 的顶点为 (3, −4)，对称轴为 x = 3。
he equation to find the y-coordinate. This gives the exact minimum or maximum value of the function.
要从标准形式求顶点，先把 x = −b / (2a) 代入方程求出 y 坐标。这样就能得到函数精确的最小值或最大值。
9. Solving Quadratic-Linear Systems | 二次与线性方程组的求解
2x + 1 and y = x + 3. Substitute the linear expression into the quadratic to eliminate y.
一些 IGCSE 题目会要求同时解一个二次方程和一个线性方程，例如 y = x² + 2x + 1 和 y = x + 3。将线性表达式代入二次方程，以消去 y。
g gives (x + 2)(x − 1) = 0, so x = −2 or x = 1. Then substitute back to find the y-values.
这样得到 x² + 2x + 1 = x + 3，化简为 x² + x − 2 = 0。因式分解得 (x + 2)(x − 1) = 0，所以 x = −2 或 x = 1。然后代回原方程求出对应的 y 值。
The number of intersection points between the line and the parabola can also be found using the discriminant of the resulting quadratic equation.
直线与抛物线的交点个数也可以通过所得一元二次方程的判别式来判断。
10. Exam-Style Applications and Common Errors | 考试型应用与常见错误
Quadratic equations often appear in word problems involving area, projectile motion, or profit. Define a variable clearly, form the equation, solve it, and check that the answer makes sense in the context.
二次方程经常出现在涉及面积、抛体运动或利润的应用题中。要清晰地设出未知数，列出方程，求解，并检查答案在题目情境中是否合理。
x, which may lose the solution x = 0. Instead, bring all terms to one side and factorise.
一个常见错误是将二次方程两边同时除以 x，这样可能会丢掉 x = 0 这个解。正确做法是把所有项移到一边，然后因式分解。
= 0 first, and do not divide by a variable expression unless you are sure it is non-zero.
另一个常见错误是在因式分解前忘记把方程整理为零。务必先写成 ax² + bx + c = 0，并且除非确定某个含变量的表达式不等于零，否则不要除以它。
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