Quadratic Equations | 一元二次方程

📚 Quadratic Equations | 一元二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, and word problems, and mastering them is essential for achieving a high grade.

一元二次方程是 IGCSE 数学中考查频率最高的考点之一，它贯穿代数、坐标几何和应用题，掌握这一知识点是取得高分的关键。

1. What Is a Quadratic Equation? | 什么是一元二次方程

A quadratic equation is a polynomial equation in which the highest power of the unknown variable is 2. Its general form is written as:

一元二次方程是未知数的最高次数为 2 的多项式方程，其一般形式可写成：

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are constants, x is the variable, and the coefficient a must not be zero — otherwise the equation is linear, not quadratic.

其中 a、b、c 为常数，x 为未知数，且二次项系数 a 不能为零——否则方程就变成了一次方程，而非二次方程。

For example, x² − 3x + 2 = 0 is quadratic, while 2x − 5 = 0 is not.

例如，x² − 3x + 2 = 0 是一元二次方程，而 2x − 5 = 0 不是。

2. Standard Form and Rearranging | 标准形式与整理方程

Before solving any quadratic equation, always rearrange it into the standard form ax² + bx + c = 0. This means moving all terms to one side of the equals sign so that the other side is zero.

在求解任何一元二次方程之前，都要先将方程整理成标准形式 ax² + bx + c = 0，即将所有项移到等号的一侧，使另一侧为零。

Consider the equation 2x² − 5x = 3. Subtracting 3 from both sides gives:

例如对于方程 2x² − 5x = 3，两边同时减去 3 得到：

2x² − 5x − 3 = 0

Once the equation is in standard form, you may choose an appropriate method to solve it, such as factorisation, completing the square, or the quadratic formula.

将方程整理为标准形式后，就可以选择合适的解法，包括因式分解法、配方法和求根公式法。

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer roots. We look for two numbers whose product equals ac and whose sum equals b.

当二次方程具有简单的整数根时，因式分解法通常是最快捷的方法。我们需要找到两个数，使它们的乘积等于 ac，和等于 b。

Solve x² − 5x + 6 = 0. We need two numbers that multiply to 6 and add to −5. These are −2 and −3:

解方程 x² − 5x + 6 = 0。我们需要找到两个数，乘积为 6，和为 −5，这两个数是 −2 和 −3：

(x − 2)(x − 3) = 0

For the product of two factors to be zero, at least one factor must be zero. Therefore:

要使两个因式的乘积为零，至少其中一个因式必须为零。因此：

x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3

Always check your solutions by substituting them back into the original equation.

务必把解代回原方程进行检验。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that works for any quadratic equation. It involves rewriting the quadratic in the form (x + p)² + q.

配方法是一种适用于任何一元二次方程的通用技巧，它的核心是将二次式改写为 (x + p)² + q 的形式。

Solve x² + 6x + 5 = 0. First, focus on the x² and 6x terms. Since (x + 3)² = x² + 6x + 9, we write:

解方程 x² + 6x + 5 = 0。首先关注 x² 项和 6x 项。由于 (x + 3)² = x² + 6x + 9，我们可以写为：

x² + 6x + 9 − 9 + 5 = 0

This simplifies to:

化简得到：

(x + 3)² − 4 = 0

(x + 3)² = 4

Taking the square root of both sides gives x + 3 = ±2, so:

两边开平方得 x + 3 = ±2，因此：

x = −1 or x = −5

If the coefficient of x² is not 1, divide the whole equation by that coefficient before completing the square.

如果 x² 的系数不为 1，应先让整个方程除以该系数，再进行配方。

5. The Quadratic Formula | 求根公式

The quadratic formula is a universal method that solves any quadratic equation, even when factorisation is difficult or impossible. For ax² + bx + c = 0:

求根公式是一种通用的解法，即使方程难以因式分解，它也能解决问题。对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / 2a

Use the formula to solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2. Substituting:

用求根公式解 2x² + 3x − 2 = 0。其中 a = 2，b = 3，c = −2。代入公式：

x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2)

x = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4

Since √25 = 5, we obtain two solutions:

由于 √25 = 5，我们得到两个解：

x = (−3 + 5) / 4 = 1/2 or x = (−3 − 5) / 4 = −2

The ± symbol indicates that we must consider both the positive and negative square root.

± 符号表示我们必须同时考虑正平方根和负平方根两种情况。

6. The Discriminant | 判别式

The expression b² − 4ac in the quadratic formula is called the discriminant, denoted by Δ. It tells us about the nature of the roots without fully solving the equation.

求根公式中的 b² − 4ac 称为判别式，记作 Δ。它可以帮助我们不解方程就判断根的性质。

Discriminant Δ = b² − 4ac	Nature of roots \| 根的性质
Δ > 0	Two distinct real roots \| 两个不相等的实数根
Δ = 0	Two equal real roots (one repeated root) \| 两个相等的实数根（一个重根）
Δ < 0	No real roots \| 没有实数根

For example, the equation x² − 4x + 4 = 0 has Δ = (−4)² − 4 × 1 × 4 = 0, so it has exactly one repeated root: x = 2.

例如，方程 x² − 4x + 4 = 0 的判别式 Δ = (−4)² − 4 × 1 × 4 = 0，因此只有一个重根：x = 2。

In IGCSE exams, questions often ask you to find the value of a parameter for which the equation has equal roots. Set Δ = 0 and solve for the parameter.

IGCSE 考试中经常会出现求参数值使方程有等根的问题，此时令 Δ = 0 并求解参数即可。

7. Graphs of Quadratic Functions | 二次函数的图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upward and has a minimum point; if a < 0, it opens downward and has a maximum point.

函数 y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上，有最小值点；当 a < 0 时，抛物线开口向下，有最大值点。

The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the graph. The x-coordinate of the vertex (turning point) is given by:

方程 ax² + bx + c = 0 的根对应图像与 x 轴的交点。顶点（驻点）的横坐标由下式给出：

x = −b / 2a

To find the y-coordinate of the vertex, substitute this x-value into the original function. The y-intercept is simply the value of c, since when x = 0, y = c.

要计算顶点的纵坐标，只需将该 x 值代回原函数即可。y 轴截距就是 c 的值，因为当 x = 0 时，y = c。

Sketching a parabola in an exam requires four key features: the roots, the vertex, the y-intercept, and the direction of opening.

在考试中画抛物线草图需要标出四个关键要素：根、顶点、y 轴截距以及开口方向。

8. Word Problems | 应用题

Quadratic equations are frequently used to model real-world problems. A typical IGCSE question involves geometric shapes or projectile motion.

一元二次方程经常用于建模实际问题。IGCSE 的典型题目涉及几何图形或抛体运动。

Consider the following problem: The length of a rectangle is 3 cm longer than its width, and the area is 40 cm². Find the dimensions.

请看下面的问题：一个长方形的长比宽长 3 cm，面积为 40 cm²，求长方形的尺寸。

Let the width be x cm. Then the length is (x + 3) cm. Since area = length × width:

设宽为 x cm，则长为 (x + 3) cm。由于面积 = 长 × 宽：

x(x + 3) = 40

x² + 3x − 40 = 0

Factorising gives (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a length cannot be negative, x = 5.

因式分解得 (x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于长度不能为负数，x = 5。

The width is 5 cm and the length is 8 cm. Always reject negative solutions when the context involves measurements.

宽为 5 cm，长为 8 cm。当实际情境涉及度量时，务必舍去负根。

9. Exam Tips and Common Mistakes | 考试技巧与常见错误

The following points will help you avoid losing marks in the IGCSE examination.

以下要点能帮助你在 IGCSE 考试中避免失分。

Always rearrange the equation into the form ax² + bx + c = 0 before attempting to solve it.

在尝试求解之前，务必先将方程整理为 ax² + bx + c = 0 的形式。
If the coefficient of x² is negative, multiply the entire equation by −1 to make it positive — this makes factorisation much easier.

如果 x² 的系数为负数，将整个方程乘以 −1 化为正数，这样因式分解会容易得多。
Show all working steps clearly. Even if your final answer is wrong, you may earn method marks.

清晰写出所有解题步骤。即使最终答案有误，你仍有可能获得方法分。
When solving word problems, reject any root that does not make sense in the context, such as negative distances or times.

在解答应用题时，要舍去在情境中不合理的根，例如负的距离或时间。
Always substitute your answers back into the original equation to verify that they are correct.

始终将答案代回原方程进行验证，以确保正确。
If factorisation fails quickly, switch to the quadratic formula rather than wasting time.

如果因式分解难以快速完成，应改用求根公式，不要浪费时间。

One common mistake is forgetting that a quadratic equation can have two solutions. When using the quadratic formula, remember to calculate both the + and − cases.

一个常见错误是忘记一元二次方程可能有两个解。使用求根公式时，务必分别计算 + 和 − 两种情况。

Quadratic equations are a fundamental topic that connects algebraic manipulation, graphing, and problem solving. With consistent practice and careful attention to the methods outlined above, you can approach any quadratic question with confidence.

一元二次方程是一个基础性考点，它将代数运算、函数图像和问题解决紧密联系起来。通过持续练习并谨慎运用上述方法，你可以自信地应对任何二次方程问题。

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Quadratic Equations | 一元二次方程