📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握 IGCSE 数学二次方程
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, graphs, and real-life modelling, so a strong understanding is essential for high marks. This article explains the key methods, common exam pitfalls, and how to apply them confidently.
二次方程是 IGCSE 数学最重要的主题之一。它们出现在代数、图像和实际建模中，因此扎实的理解对于取得高分至关重要。本文讲解关键方法、常见考试陷阱以及如何自信地应用它们。
1. What Is a Quadratic Equation? | 什么是二次方程？
est power. The general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
二次方程是一个二次多项式方程。它包含一个最高次数为 2 的变量。一般形式为 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。
ax² + bx + c = 0, a ≠ 0
If a were equal to 0, the equation would become linear, not quadratic. Recognising this standard form is the first step in problem solving.
如果 a 等于 0，方程就变成一次方程，而不是二次方程。识别这种标准形式是解题的第一步。
2. Standard Form and Coefficients | 标准形式与系数
values a, b and c are called coefficients. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term.
在标准形式 ax² + bx + c = 0 中，a、b、c 的值称为系数。系数 a 是二次项系数，b 是一次项系数，c 是常数项。
o that one side equals 0. For example, 3x² = 5x - 2 should be written as 3x² - 5x + 2 = 0.
在求解之前，始终将方程整理为一边等于 0。例如，3x² = 5x - 2 应写成 3x² - 5x + 2 = 0。
3x² - 5x + 2 = 0
3. Solving by Factorising | 因式分解法
Factorising is often the fastest method when the quadratic can be written as the product of two brackets. We look for two numbers that multiply to give ac and add to give b.
当二次方程可以写成两个括号的乘积时，因式分解通常是最快的方法。我们需要找到两个数，它们相乘等于 ac，相加等于 b。
12 = 0, the numbers 3 and 4 multiply to 12 and add to 7, so it becomes (x + 3)(x + 4) = 0.
对于 x² + 7x + 12 = 0，数字 3 和 4 相乘得 12，相加得 7，因此它变为 (x + 3)(x + 4) = 0。
(x + 3)(x + 4) = 0 → x = -3 or x = -4
racket to zero gives x = -3 or x = -4. Always present your final answer as a solution set.
令每个括号等于零，得到 x = -3 或 x = -4。始终将最终答案表示为解集。
4. The Quadratic Formula | 求根公式
The quadratic formula solves any quadratic equation, even when factorising is difficult. It is derived from completing the square and is given by:
求根公式可以求解任何二次方程，即使因式分解很困难。它由配方法推导得出，公式为：
x = (-b ± √b² - 4ac) / 2a
n substitute them into the formula. For 2x² + 3x - 2 = 0, we have a = 2, b = 3 and c = -2.
仔细确定 a、b、c，然后将它们代入公式。对于 2x² + 3x - 2 = 0，我们有 a = 2，b = 3，c = -2。
This method always gives the roots, including irrational or non-integer answers, so it is very reliable in exams.
这种方法总能求出根，包括无理数或非整数答案，因此在考试中非常可靠。
5. Completing the Square | 配方法
Completing the square rewrites the quadratic in the form a(x + p)² + q. This form reveals the turning point of the graph and is useful for solving and sketching.
配方法将二次式改写为 a(x + p)² + q 的形式。这种形式可以揭示图像的顶点，对求解和画图很有用。
take half of 6, which is 3, square it to get 9, and write (x + 3)² - 9 + 5 = (x + 3)² - 4.
对于 x² + 6x + 5，取 6 的一半为 3，平方得 9，写成 (x + 3)² - 9 + 5 = (x + 3)² - 4。
x² + 6x + 5 = (x + 3)² - 4
uation x² + 6x + 5 = 0 becomes (x + 3)² - 4 = 0, giving (x + 3)² = 4, so x = -1 or x = -5.
方程 x² + 6x + 5 = 0 变为 (x + 3)² - 4 = 0，得到 (x + 3)² = 4，因此 x = -1 或 x = -5。
6. The Discriminant | 判别式
quadratic formula: D = b² - 4ac. It tells us how many real roots a quadratic equation has.
判别式是求根公式平方根内的表达式：D = b² - 4ac。它告诉我们二次方程有多少个实数根。
D = b² - 4ac
real roots. If D = 0, there is one repeated real root. If D < 0, there are no real roots.
如果 D > 0，则有两个不同的实数根；如果 D = 0，则有一个重根；如果 D < 0，则没有实数根。
This concept is often tested without asking you to solve the equation. Just calculate the discriminant and state the nature of the roots.
这个概念经常在考试中不要求解方程，只需计算判别式并说明根的性质。
7. Graphs of Quadratic Functions | 二次函数图像
duces a parabola. If a > 0, the parabola opens upwards with a minimum point. If a < 0, it opens downwards with a maximum point.
二次函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，有一个最低点；如果 a < 0，开口向下，有一个最高点。
the square or using x = -b / 2a. The y-intercept is always at c, because y = c when x = 0.
顶点可以通过配方法或使用 x = -b / 2a 求得。y 轴截距始终在 c 处，因为当 x = 0 时 y = c。
x = -b / 2a
Sketching the graph requires the roots, the turning point and the y-intercept. Label these clearly to earn full marks.
画出图像需要根、顶点和 y 轴截距。清晰地标注这些要素才能获得满分。
8. Roots and Intercepts | 根与截距
The roots of a quadratic equation are the x-values where the graph crosses the x-axis. They are also called solutions or x-intercepts.
二次方程的根是图像与 x 轴相交处的 x 值。它们也被称为解或 x 轴截距。
uadratic equation ax² + bx + c = 0. To find the y-intercept, set x = 0 and evaluate y = c.
要求根，令 y = 0 并解二次方程 ax² + bx + c = 0。要求 y 轴截距，令 x = 0 并计算 y = c。
e x-axis at only one point, the quadratic has a repeated root, which corresponds to D = 0.
如果图像仅在一个点与 x 轴相切，则该二次方程有一个重根，对应 D = 0。
Roots can be read from a graph, but exam questions often ask for exact values, so use algebra to confirm.
根可以从图像中读取，但考试题目通常要求精确值，因此用代数方法进行确认。
9. Applications and Word Problems | 应用题与文字题
Quadratic equations model many real-life situations such as projectile motion, areas of rectangles, and profit functions. Translate the words into an equation before solving.
二次方程可以模拟许多现实生活中的情形，如抛体运动、矩形面积和利润函数。在求解之前，先将文字转化为方程。
idth x, with area 40. The equation is x(x + 3) = 40, which simplifies to x² + 3x - 40 = 0.
例如，一个矩形的长为 x + 3，宽为 x，面积为 40。方程为 x(x + 3) = 40，化简为 x² + 3x - 40 = 0。
x(x + 3) = 40 → x² + 3x - 40 = 0
Solve the equation and discard any negative solution that is not physically possible. Then answer the original question with correct units
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