📚 G-2 Workbook Exercise 100: Mastering Quadratic Equations | G-2练习册第100题：掌握二次方程
This article walks you through the key skills needed to solve quadratic equations, with a special focus on the type of problem found in Exercise 100 of the G-2 workbook. You will learn how to factorise, use the quadratic formula, complete the square, and interpret graphs — all essential for your IGCSE exam.
本文带你掌握解二次方程所需的关键技能，并特别关注G-2练习册第100题中出现的题型。你将学习如何因式分解、使用求根公式、配方法以及解读图像——这些都是IGCSE考试的核心内容。
1. Understanding Quadratic Equations | 理解二次方程
A quadratic equation is any equation that can be written in the form ax2 + bx + c = 0 , where a , b , and c are constants and a ≠ 0 . The highest power of the variable is 2, which gives the equation its name 'quadratic'.
二次方程是指可以写成 ax2 + bx + c = 0 形式的方程，其中 a 、 b 、 c 为常数，且 a ≠ 0 。变量的最高次数为2，因此得名“二次”。
For example, 2x2 - 3x + 1 = 0 is a quadratic equation, while x3 + x = 0 is not, because the highest power is 3.
例如， 2x2 - 3x + 1 = 0 是二次方程，而 x3 + x = 0 不是，因为最高次数是3。
2. Standard Form and Key Features | 标准形式与关键特征
The standard form of a quadratic equation is ax2 + bx + c = 0 . The value of a determines the shape of the graph: if a > 0 , the graph is a U-shaped parabola; if a < 0 , it is an upside-down U (n-shaped).
二次方程的标准形式是 ax2 + bx + c = 0 。系数 a 决定图像形状：若 a > 0 ，图像是U形抛物线；若 a < 0 ，则是倒U形（n形）。
The constant c represents the y-intercept, because when x = 0 , y = c . The roots (solutions) are the x-values where the graph crosses the x-axis.
常数 c 表示y轴截距，因为当 x = 0 时， y = c 。根（解）是图像与x轴交点的x值。
3. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when the quadratic has simple integer factors. The idea is to write ax2 + bx + c as a product of two linear factors.
当二次式具有简单的整数因子时，因式分解通常是最快的方法。思路是把 ax2 + bx + c 写成两个一次因式的乘积。
For example, solve x2 - 5x + 6 = 0 . We look for two numbers that multiply to 6 and add to -5. These are -2 and -3, so:
例如，解 x2 - 5x + 6 = 0 。我们需要找两个数，乘积为6，和为-5。这两个数是-2和-3，因此：
(x - 2)(x - 3) = 0
Then set each factor to zero: x - 2 = 0 or x - 3 = 0 , giving x = 2 or x = 3 .
然后令每个因式为零： x - 2 = 0 或 x - 3 = 0 ，得 x = 2 或 x = 3 。
4. Factorising Non-Monic Quadratics | 非首一二次式的因式分解
When a ≠ 1 , factorisation requires extra care. For example, solve 2x2 + 7x + 3 = 0 .
当 a ≠ 1 时，因式分解需要格外小心。例如，解 2x2 + 7x + 3 = 0 。
We look for factors of 2 × 3 = 6 that add to 7. The pair is 1 and 6. Rewrite the middle term:
我们找 2 × 3 = 6 的因数，使其和为7。这一对数是1和6。重写中间项：
2x² + 1x + 6x + 3 = 0
Factor by grouping: x(2x + 1) + 3(2x + 1) = 0 , so (x + 3)(2x + 1) = 0 . Hence x = -3 or x = -12 .
分组因式分解： x(2x + 1) + 3(2x + 1) = 0 ，所以 (x + 3)(2x + 1) = 0 。因此 x = -3 或 x = -12 。
5. Solving by the Quadratic Formula | 公式法求解
The quadratic formula solves any quadratic equation, even when factorisation is difficult or impossible. For ax2 + bx + c = 0 , the formula is:
求根公式可以解任何二次方程，即使因式分解困难或不可能。对于 ax2 + bx + c = 0 ，公式为：
x = (−b ± √b² − 4ac) / (2a)
For example, solve 2x2 - 4x - 3 = 0 . Here a = 2 , b = -4 , c = -3 . Substitute:
例如，解 2x2 - 4x - 3 = 0 。这里 a = 2 ， b = -4 ， c = -3 。代入：
x = (4 ± √16 + 24) / 4 = (4 ± √40) / 4
So x = 2 + √102 or x = 2 - √102 . These are exact surd answers.
所以 x = 2 + √102 或 x = 2 - √102 。这些是精确的根式答案。
6. Solving by Completing the Square | 配方法求解
Completing the square changes the quadratic into the form a(x + p)2 + q . This is especially useful for finding turning points and solving equations.
配方法将二次式变形为 a(x + p)2 + q 的形式。这对于求顶点和求解方程特别有用。
For example, solve x2 + 6x + 2 = 0 . Take half of 6, square it (9), and rewrite:
例如，解 x2 + 6x + 2 = 0 。取6的一半，平方得9，重写：
(x + 3)² − 9 + 2 = 0 → (x + 3)² − 7 = 0
Then (x + 3)² = 7 , so x + 3 = ±√7 , giving x = −3 ± √7 .
然后 (x + 3)² = 7 ，所以 x + 3 = ±√7 ，得 x = −3 ± √7 。
7. The Discriminant | 判别式
The discriminant is the expression b2 - 4ac inside the square root of the quadratic formula. It tells us how many real roots a quadratic equation has.
判别式是求根公式中根号内的表达式 b2 - 4ac 。它告诉我们二次方程有多少个实数根。
- If b2 - 4ac > 0 , there are two distinct real roots.
- 如果 b2 - 4ac > 0 ，有两个不同的实数根。
- If b2 - 4ac = 0 , there is one repeated real root.
- 如果 b2 - 4ac = 0 ，有一个相等的实数根（重根）。
- If b2 - 4ac < 0 , there are no real roots.
- 如果 b2 - 4ac < 0 ，没有实数根。
For example, the equation x2 + 2x + 5 = 0 has discriminant 22 - 4(1)(5) = 4 - 20 = -16 < 0 , so it has no real roots.
例如，方程 x2 + 2x + 5 = 0 的判别式为 22 - 4(1)(5) = 4 - 20 = -16 < 0 ，因此没有实数根。
8. Graphing Quadratic Functions | 二次函数图像
A quadratic function y = ax2 + bx + c always produces a parabola. The vertex (turning point) can be found by completing the square: if y = a(x - h)2 + k , then the vertex is at (h, k).
二次函数 y = ax2 + bx + c 的图像总是抛物线。顶点（转向点）可通过配方法找到：若 y = a(x - h)2 + k ，则顶点在 (h, k)。
For y = x2 - 4x + 1 , complete the square: (x - 2)2 - 3 . Thus the vertex is (2, -3), and the axis of symmetry is x = 2 .
对于 y = x2 - 4x + 1 ，配方得 (x - 2)2 - 3 。因此顶点是 (2, -3)，对称轴是 x = 2 。
The y-intercept is at (0, c) . The x-intercepts are the roots of the equation ax2 + bx + c = 0 .
y轴截距是 (0, c) 。x轴截距是方程 ax2 + bx + c = 0 的根。
9. Solving Quadratic Inequalities | 解二次不等式
Quadratic inequalities require the same root-finding techniques, but the solution is a range of x-values.
二次不等式需要相同的求根技巧，但解是一个x值的范围。
For example, solve x2 - x - 6 < 0 . Factorise: (x - 3)(x + 2) < 0 . The roots are x = 3 and x = -2 .
例如，解 x2 - x - 6 < 0 。因式分解： (x - 3)(x + 2) < 0 。根为 x = 3 和 x = -2 。
Sketch the parabola or test intervals. The inequality holds between the roots, so the solution is -2 < x < 3 .
画出抛物线或测试区间。不等式在两根之间成立，所以解为 -2 < x < 3 。
10. Workbook 100: Worked Example | 练习册第100题：例题精讲
Let's solve a problem typical of Exercise 100 in the G-2 workbook. The question asks you to solve 3x2 - 5x - 2 = 0 by factorisation.
我们来解一道G-2练习册第100题的典型题目。题目要求用因式分解法解 3x2 - 5x - 2 = 0 。
First, multiply a × c = 3 × (-2) = -6 . We need two numbers that multiply to -6 and add to -5: these are -6 and 1.
首先，计算 a × c = 3 × (-2) = -6 。我们需要两个数，乘积为-6，和为-5：这两个数是-6和1。
Rewrite: 3x2 - 6x + x - 2 = 0 . Then group: 3x(x - 2) + 1(x - 2) = 0 .
重写： 3x2 - 6x + x - 2 = 0 。然后分组： 3x(x - 2) + 1(x - 2) = 0 。
Factor: (3x + 1)(x - 2) = 0 . Hence x = -13 or x = 2 .
提取公因式： (3x + 1)(x - 2) = 0 。因此 x = -13 或 x = 2 。
11. Common Mistakes to Avoid | 常见错误与规避
Many students make the same mistakes when solving quadratics. Here are the most frequent ones and how to avoid them.
许多学生在解二次方程时会犯同样的错误。以下是最常见的错误及避免方法。
- Forgetting to rearrange the equation into the form ax2 + bx + c = 0 before factorising.
- 在因式分解前忘记将方程整理成 ax2 + bx + c = 0 的形式。
- Dropping the negative sign when b or c is negative — always double-check signs.
- 当 b 或 c 为负时弄错符号——务必仔细检查正负号。
- Using the quadratic formula without simplifying the square root fully.
- 使用求根公式时没有完全化简根式。
- Confusing the roots with the vertex when sketching graphs.
- 画图时将根与顶点混淆。
12. Summary and Final Tips | 总结与考前建议
To master quadratic equations, you need to be fluent in all three solving methods: factorisation, the quadratic formula, and completing the square. Each has its place, and the exam may ask you to use a specific one.
要掌握二次方程，你需要熟练三种解法：因式分解、求根公式和配方法。每种方法各有用途，考试可能要求使用特定方法。
Always check whether the equation is in standard form first. If asked for exact answers, leave surds in the form √k ; if asked for decimal answers, use your calculator carefully.
务必先检查方程是否为标准形式。如果要求精确答案，保留 √k 形式的根式；如果要求小数答案，请小心使用计算器。
Practice with the G-2 workbook, especially Exercise 100, to build speed and confidence. Good luck with your revision!
请使用G-2练习册进行练习，特别是第100题，以提升速度和信心。祝复习顺利！
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