📚 IGCSE Maths Topic 2: Quadratic Equations (T-2-1055) | IGCSE 数学专题 2：二次方程（T-2-1055）
Quadratic equations are one of the most tested algebra topics in IGCSE Mathematics. This revision guide covers solving methods, the discriminant, graphing, inequalities and real-life applications, with clear bilingual explanations and worked examples for exam success.
二次方程是 IGCSE 数学中最常考的代数主题之一。本复习指南涵盖求解方法、判别式、图像、不等式和实际应用，并提供清晰的中英双语讲解和例题，帮助你在考试中取得成功。
1. What Is a Quadratic Equation? | 什么是二次方程？
stants and a ≠ 0. The highest power of the variable x is 2, so the graph is always a parabola.
二次方程是任何可以写成标准形式 a x² + b x + c = 0 的方程，其中 a、b、c 为常数且 a ≠ 0。变量 x 的最高次数是 2，因此它的图像总是一条抛物线。
c equations. The equation x³ + 2x − 1 = 0 is not quadratic because the highest power is 3.
例如，x² + 5x + 6 = 0、2x² − 3x − 2 = 0 和 x² − 9 = 0 都是二次方程。方程 x³ + 2x − 1 = 0 不是二次方程，因为最高次数是 3。
Quadratic equations appear in many IGCSE questions, from simple factorisation to word problems involving area or projectile motion. Recognising the standard form immediately will save time in the exam.
二次方程出现在许多 IGCSE 考题中，从简单的因式分解到涉及面积或抛体运动的文字题。能够立即识别标准形式将帮助你在考试中节省时间。
2. Standard Form and Key Terms | 标准形式与关键术语
called the leading coefficient, b is the linear coefficient, and c is the constant term. The value a cannot be zero, otherwise the equation becomes linear.
在标准形式 a x² + b x + c = 0 中，a 称为二次项系数，b 称为一次项系数，c 称为常数项。a 不能为零，否则方程就变成一次方程。
Solutions of a quadratic equation are called roots, zeros or x-intercepts. A root is a value of x that makes the equation true. A quadratic equation can have two real roots, one repeated real root, or no real roots depending on the discriminant.
二次方程的解称为根、零点或 x 轴截距。根是使方程成立的 x 值。根据判别式的不同，二次方程可能有两个实根、一个重根或没有实根。
either value gives zero. These roots are also the points where the graph crosses the x-axis.
例如，方程 x² + x − 2 = 0 的根为 x = 1 和 x = −2，因为代入任一值都使方程等于零。这些根同时也是图像与 x 轴的交点。
| Term | Symbol | 中文 | Meaning |
| Leading coefficient | a | 二次项系数 | Determines the shape and direction of the parabola |
| Linear coefficient | b | 一次项系数 | Affects the position of the axis of symmetry |
| Constant term | c | 常数项 | The y-intercept of the graph |
3. Solving by Factorisation | 用因式分解法求解
Factorisation is usually the fastest method when the roots are integers or simple fractions. First write the equation in standard form, then factorise the quadratic expression into two brackets.
当根为整数或简单分数时，因式分解通常是最快的方法。首先把方程写成标准形式，然后把二次表达式分解成两个括号。
to 5. These are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3) = 0. Therefore x = −2 or x = −3.
对于 x² + 5x + 6 = 0，我们需要两个数，它们的乘积为 6，和为 5。这两个数是 2 和 3，所以 x² + 5x + 6 = (x + 2)(x + 3) = 0。因此 x = −2 或 x = −3。
r x² − 9 = 0, this is a difference of two squares: (x − 3)(x + 3) = 0, so x = 3 or x = −3.
对于 2x² − 3x − 2 = 0，因式分解形式为 (2x + 1)(x − 2) = 0，得到 x = −½ 或 x = 2。对于 x² − 9 = 0，这是平方差：(x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。
- Always move all terms to one side so the equation equals zero. | 始终把所有项移到一边，使方程等于零。
- Check by expanding the brackets or substituting the roots back. | 通过展开括号或代回根来检验。
- If factorisation is not obvious, use completing the square or the quadratic formula. | 如果因式分解不明显，使用配方法或求根公式。
4. Completing the Square | 配方法
ethod is useful for finding the vertex of a parabola and for solving equations that do not factorise easily.
配方法把二次式改写为 (x + p)² + q = 0 的形式。这个方法对于求抛物线的顶点以及求解不易因式分解的方程非常有用。
0, take half of b, square it, and add and subtract it. This gives (x + b/2)² = (b/2)² − c.
对于 x² + b x + c = 0，取 b 的一半，平方后加减它。得到 (x + b/2)² = (b/2)² − c。
² − 9 + 5 = 0, so (x + 3)² = 4. Taking square roots gives x + 3 = ±2, so x = −1 or x = −5.
例如，x² + 6x + 5 = 0 变为 (x + 3)² − 9 + 5 = 0，所以 (x + 3)² = 4。开平方得到 x + 3 = ±2，因此 x = −1 或 x = −5。
(x + b/2)² = (b/2)² − c
5. The Quadratic Formula | 二次公式（求根公式）
ic equation, even when factorisation is difficult. For a x² + b x + c = 0, the formula is:
二次公式可以求解任何二次方程，即使因式分解很困难。对于 a x² + b x + c = 0，公式为：
x = (−b ± √b² − 4ac) / (2a)
c = (−3)² − 4 × 2 × (−2) = 9 + 16 = 25. Therefore x = (3 ± 5) / 4, giving x = 2 or x = −½.
对于 2x² − 3x − 2 = 0，a = 2，b = −3，c = −2。判别式为 b² − 4ac = (−3)² − 4 × 2 × (−2) = 9 + 16 = 25。因此 x = (3 ± 5) / 4，得到 x = 2 或 x = −½。
- Write down a, b and c carefully, including negative signs. | 仔细写出 a、b、c，包括负号。
- Calculate the discriminant first to predict the number of roots. | 先计算判别式以预判根的个数。
- Simplify the surd if the answer is not a whole number. | 如果答案不是整数，化简根式。
6. The Discriminant and Number of Roots | 判别式与根的个数
Δ = b² − 4ac. It tells you how many real roots the equation has without solving it fully.
判别式是二次公式中根号下的表达式：Δ = b² − 4ac。它可以在不完全求解的情况下告诉你方程有多少个实根。
| Discriminant | Number of real roots | 中文说明 |
| Δ > 0 | Two distinct real roots | 两个不同的实根，图像与 x 轴有两个交点 |
| Δ = 0 | One repeated real root | 一个重根，图像与 x 轴相切 |
| Δ < 0 | No real roots | 没有实根，图像与 x 轴不相交 |
oots. The equation x² + 2x + 1 = 0 has Δ = 4 − 4 = 0, so it has one repeated root, x = −1.
例如，x² + x − 2 = 0 的 Δ = 1² − 4 × 1 × (−2) = 9 > 0，所以有两个实根。方程 x² + 2x + 1 = 0 的 Δ = 4 − 4 = 0，所以有一个重根 x = −1。
7. Graphs of Quadratic Functions | 二次函数图像
if a < 0, it opens downwards. The axis of symmetry is the vertical line x = −b / (2a).
y = a x² + b x + c 的图像是一条抛物线。如果 a > 0，开口向上；如果 a < 0，开口向下。对称轴是竖直线 x = −b / (2a)。
The vertex is the turning point of the parabola. Its x-coordinate is −b / (2a), and its y-coordinate can be found by substituting this value into the equation. The y-intercept is c, and the x-intercepts are the roots.
顶点是抛物线的转折点。它的横坐标是 −b / (2a)，纵坐标可以通过把这个值代入方程得到。y 轴截距是 c，x 轴截距是方程的根。
−2, the axis of symmetry is x = −½, and the vertex is (−½, −2.25). The y-intercept is −2.
对于 y = x² + x − 2，根为 x = 1 和 x = −2，对称轴为 x = −½，顶点为 (−½, −2.25)。y 轴截距为 −2。
8. Quadratic Inequalities | 二次不等式
A quadratic inequality compares a quadratic expression with zero, such as x² + x − 2 > 0 or x² + x − 2 < 0. First find the critical values by solving the corresponding equation.
二次不等式是把二次表达式与零进行比较，例如 x² + x − 2 > 0 或 x² + x − 2 < 0。首先通过解对应的方程找到临界值。
re x = 1 and x = −2. The parabola opens upwards, so the solution is x < −2 or x > 1.
对于 x² + x − 2 > 0，因式分解为 (x − 1)(x + 2) > 0。临界值是 x = 1 和 x = −2。抛物线开口向上，所以解为 x < −2 或 x > 1。
For x² + x − 2 < 0, the same critical values apply, but the solution is the interval between them: −2 < x < 1. Always sketch the parabola to check the direction of the inequality.
对于 x² + x − 2 < 0，临界值相同，但解是它们之间的区间：−2 < x < 1。始终画出抛物线草图来检查不等号的方向。
9. Simultaneous Equations with a Quadratic | 与二次方程联立的方程组
Some IGCSE problems involve a linear equation and a quadratic equation. Substitute the linear equation into the quadratic equation to form a single quadratic equation in one variable.
有些 IGCSE 题目涉及一个一次方程和一个二次方程。把一次方程代入二次方程，形成一个单变量二次方程。
s x² = x + 2, so x² − x − 2 = 0. Factorising gives (x − 2)(x + 1) = 0, so x = 2 or x = −1.
例如，解 y = x² 和 y = x + 2。代入得到 x² = x + 2，所以 x² − x − 2 = 0。因式分解得到 (x − 2)(x + 1) = 0，所以 x = 2 或 x = −1。
The solutions are (2, 4) and (−1, 1). Always give both x and y values as coordinate pairs.
当 x = 2 时，y = 4。当 x = −1 时，y = 1。解为 (2, 4) 和 (−1, 1)。始终把 x 和 y 的值作为坐标对给出。
10. Word Problems and Real-Life Applications | 文字题与实际应用
width and an area of 40 cm². Let the width be w, then w(w + 3) = 40, so w² + 3w − 40 = 0.
二次方程常用于面积问题。例如，一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 w，则 w(w + 3) = 40，所以 w² + 3w − 40 = 0。
, w = 5 cm. The length is 8 cm. Always reject solutions that do not make sense in context.
因式分解得到 (w + 8)(w − 5) = 0，所以 w = −8 或 w = 5。由于宽度不能为负，w = 5 cm。长为 8 cm。始终舍去在情境中不合理的解。
20t − 5t², it hits the ground when h = 0, so 5t(4 − t) = 0, giving t = 0 or t = 4 seconds.
抛体运动也会用到二次方程。如果一个球以高度 h = 20t − 5t² 向上抛出，当 h = 0 时落地，所以 5t(4 − t) = 0，得到 t = 0 或 t = 4 秒。
11. Common Exam Mistakes and Revision Tips | 常见错误与复习建议
Many marks are lost through careless mistakes rather than lack of understanding. One common error is forgetting to rearrange the equation into standard form before factorising.
许多失分是由于粗心错误，而不是因为不理解。一个常见错误是在因式分解前忘记把方程整理成标准形式。
- Do not divide by x if x could be zero; factorise instead. | 如果 x 可能为零，不要两边除以 x；应当因式分解。
- Check the signs when using the quadratic formula, especially if b is negative. | 使用求根公式时检查符号，尤其是 b 为负数时。
- For inequalities, sketch the parabola or use a number line to avoid reversing the interval. | 解不等式时画出抛物线或使用数轴，避免把区间写反。
- In word problems, reject negative or unrealistic roots. | 在文字题中，舍去负数或不合理的根。
- Practise past-paper questions under timed conditions. | 在限时条件下练习历年真题。
Remember that the discriminant only tells you the number of real roots; it does not give the roots themselves. Use it as a quick check before spending time on a full solution.
记住，判别式只告诉你实根的个数，它并不给出根本身。把它当作快速检查，再花时间完整求解。
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