G-2 Workbook Exercise 100: Mastering Quadratic Equations | G-2练习册第100题：掌握二次方程

📚 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 \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \neq 0 \). 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 \( ax^2 + bx + c = 0 \) 形式的方程，其中 \( a \)、\( b \)、\( c \) 为常数，且 \( a \neq 0 \)。变量的最高次数为2，因此得名“二次”。

For example, \( 2x^2 – 3x + 1 = 0 \) is a quadratic equation, while \( x^3 + x = 0 \) is not, because the highest power is 3.

例如，\( 2x^2 – 3x + 1 = 0 \) 是二次方程，而 \( x^3 + x = 0 \) 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 \( ax^2 + bx + c = 0 \). 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 \( ax^2 + bx + c = 0 \)。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 \( ax^2 + bx + c \) 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 \( ax^2 + bx + c \) 写成两个一次因式的乘积。

For example, solve \( x^2 – 5x + 6 = 0 \). We look for two numbers that multiply to 6 and add to -5. These are -2 and -3, so:

例如，解 \( x^2 – 5x + 6 = 0 \)。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 \( a \neq 1 \), factorisation requires extra care. For example, solve \( 2x^2 + 7x + 3 = 0 \).

当 \( a \neq 1 \) 时，因式分解需要格外小心。例如，解 \( 2x^2 + 7x + 3 = 0 \)。

We look for factors of \( 2 \times 3 = 6 \) that add to 7. The pair is 1 and 6. Rewrite the middle term:

我们找 \( 2 \times 3 = 6 \) 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 \( x = -\frac{1}{2} \).

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 \( x = -\frac{1}{2} \)。

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 \( ax^2 + bx + c = 0 \), 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 \( ax^2 + bx + c = 0 \)，公式为：

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve \( 2x^2 – 4x – 3 = 0 \). Here \( a = 2 \), \( b = -4 \), \( c = -3 \). Substitute:

例如，解 \( 2x^2 – 4x – 3 = 0 \)。这里 \( a = 2 \)，\( b = -4 \)，\( c = -3 \)。代入：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

So \( x = \frac{2 + \sqrt{10}}{2} \) or \( x = \frac{2 – \sqrt{10}}{2} \). These are exact surd answers.

所以 \( x = \frac{2 + \sqrt{10}}{2} \) 或 \( x = \frac{2 – \sqrt{10}}{2} \)。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 \( x^2 + 6x + 2 = 0 \). Take half of 6, square it (9), and rewrite:

例如，解 \( x^2 + 6x + 2 = 0 \)。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 \( b^2 – 4ac \) 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 \( b^2 – 4ac \)。它告诉我们二次方程有多少个实数根。

If \( b^2 – 4ac > 0 \), there are two distinct real roots.
如果 \( b^2 – 4ac > 0 \)，有两个不同的实数根。
If \( b^2 – 4ac = 0 \), there is one repeated real root.
如果 \( b^2 – 4ac = 0 \)，有一个相等的实数根（重根）。
If \( b^2 – 4ac < 0 \), there are no real roots.
如果 \( b^2 – 4ac < 0 \)，没有实数根。

For example, the equation \( x^2 + 2x + 5 = 0 \) has discriminant \( 2^2 – 4(1)(5) = 4 – 20 = -16 < 0 \), so it has no real roots.

例如，方程 \( x^2 + 2x + 5 = 0 \) 的判别式为 \( 2^2 – 4(1)(5) = 4 – 20 = -16 < 0 \)，因此没有实数根。

8. Graphing Quadratic Functions | 二次函数图像

A quadratic function \( y = ax^2 + bx + c \) 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 \( y = ax^2 + bx + c \) 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 \( y = x^2 – 4x + 1 \), complete the square: \( (x – 2)^2 – 3 \). Thus the vertex is \((2, -3)\), and the axis of symmetry is \( x = 2 \).

对于 \( y = x^2 – 4x + 1 \)，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 \( ax^2 + bx + c = 0 \).

y轴截距是 \( (0, c) \)。x轴截距是方程 \( ax^2 + bx + c = 0 \) 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 \( x^2 – x – 6 < 0 \). Factorise: \( (x - 3)(x + 2) < 0 \). The roots are \( x = 3 \) and \( x = -2 \).

例如，解 \( x^2 – x – 6 < 0 \)。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 \( 3x^2 – 5x – 2 = 0 \) 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 \( 3x^2 – 5x – 2 = 0 \)。

First, multiply \( a \times c = 3 \times (-2) = -6 \). We need two numbers that multiply to -6 and add to -5: these are -6 and 1.

首先，计算 \( a \times c = 3 \times (-2) = -6 \)。我们需要两个数，乘积为-6，和为-5：这两个数是-6和1。

Rewrite: \( 3x^2 – 6x + x – 2 = 0 \). Then group: \( 3x(x – 2) + 1(x – 2) = 0 \).

重写：\( 3x^2 – 6x + x – 2 = 0 \)。然后分组：\( 3x(x – 2) + 1(x – 2) = 0 \)。

Factor: \( (3x + 1)(x – 2) = 0 \). Hence \( x = -\frac{1}{3} \) or \( x = 2 \).

提取公因式：\( (3x + 1)(x – 2) = 0 \)。因此 \( x = -\frac{1}{3} \) 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 \( ax^2 + bx + c = 0 \) before factorising.
在因式分解前忘记将方程整理成 \( ax^2 + bx + c = 0 \) 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 \( \sqrt{k} \); 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 \( \sqrt{k} \) 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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G-2 Workbook Exercise 100: Mastering Quadratic Equations | G-2练习册第100题：掌握二次方程