Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in many exam questions, from simple factorisation to word problems and graph sketching. This article explains the key methods for solving quadratic equations, with clear examples and common pitfalls to avoid.

二次方程是 IGCSE 数学中最重要的考点之一。它出现在许多考试题目中，从简单的因式分解到应用题和函数图像。本文将解释求解二次方程的主要方法，并通过清晰的例子帮助你避开常见错误。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is a polynomial equation of degree 2. The highest power of the unknown variable is 2. In general, it can be written in the form:

二次方程是最高次数为 2 的多项式方程。未知数的最高次幂是 2。它的一般形式可以写成：

ax² + bx + c = 0

where a, b and c are real numbers, and a ≠ 0.

其中 a、b、c 是实数，且 a ≠ 0。

If a = 0, the equation becomes linear, not quadratic.

如果 a = 0，方程就变成了线性方程，而不是二次方程。

2. Standard Form and Key Terms | 标准形式与关键术语

Before solving a quadratic equation, you should rearrange it into standard form. This means moving all terms to one side so that the other side is 0.

在求解二次方程之前，你应当先将它整理成标准形式。这意味着把所有项移到等号一边，使另一边为 0。

For example, x² + 3x = 4 must be rewritten as x² + 3x − 4 = 0.

例如，x² + 3x = 4 必须改写为 x² + 3x − 4 = 0。

Quadratic term 二次项: ax²
Linear term 一次项: bx
Constant term 常数项: c

When you solve a quadratic equation, you are finding the values of x that make the equation true. These values are called roots or solutions.

当你求解二次方程时，你是在寻找使方程成立的 x 的值。这些值称为方程的根或解。

3. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the equation has simple integer roots. The idea is to write the quadratic as a product of two linear factors.

当方程具有简单的整数根时，因式分解往往是最快的方法。其思想是将二次式写成两个一次因式的乘积。

Example 1: Solve x² + 5x + 6 = 0.

例 1：解 x² + 5x + 6 = 0。

We look for two numbers that multiply to give 6 and add to give 5. These numbers are 2 and 3.

我们寻找两个数，它们相乘得 6，相加得 5。这两个数是 2 和 3。

x² + 5x + 6 = (x + 2)(x + 3) = 0

Therefore, x + 2 = 0 or x + 3 = 0, so x = −2 or x = −3.

因此，x + 2 = 0 或 x + 3 = 0，所以 x = −2 或 x = −3。

Example 2: Solve 2x² − 5x − 3 = 0.

例 2：解 2x² − 5x − 3 = 0。

Here we must consider the coefficient of x². The factors are (2x + 1)(x − 3).

这里需要关注 x² 的系数。因式是 (2x + 1)(x − 3)。

Check: (2x + 1)(x − 3) = 2x² − 6x + x − 3 = 2x² − 5x − 3.

检验：(2x + 1)(x − 3) = 2x² − 6x + x − 3 = 2x² − 5x − 3。

So x = −1/2 or x = 3.

所以 x = −1/2 或 x = 3。

4. Solving by Completing the Square | 配方法

Completing the square is a method that works for any quadratic equation, even when factorisation is difficult. It rewrites the equation in the form (x + p)² = q.

配方法对任何二次方程都适用，即使在因式分解困难时也能使用。它将方程改写为 (x + p)² = q 的形式。

Example: Solve x² − 6x + 4 = 0 by completing the square.

例：用配方法解 x² − 6x + 4 = 0。

Step 1: Move the constant term to the right side.

步骤 1：把常数项移到右边。

x² − 6x = −4

Step 2: Add (half of −6)² = 9 to both sides.

步骤 2：两边加上 (−6 的一半)² = 9。

x² − 6x + 9 = 5

Step 3: Write the left side as a perfect square.

步骤 3：将左边写成完全平方。

(x − 3)² = 5

Step 4: Take the square root of both sides.

步骤 4：两边开平方根。

x − 3 = ±√5

Therefore, x = 3 + √5 or x = 3 − √5.

因此，x = 3 + √5 或 x = 3 − √5。

5. The Quadratic Formula | 二次求根公式

The quadratic formula is the most reliable method. It works for every quadratic equation, no matter how difficult the numbers are.

二次求根公式是最可靠的方法。无论数字多复杂，它对所有二次方程都适用。

For ax² + bx + c = 0:

对于 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / 2a

Example: Solve 2x² + 3x − 5 = 0 using the formula.

例：用公式解 2x² + 3x − 5 = 0。

Here a = 2, b = 3, c = −5. Substitute into the formula:

这里 a = 2，b = 3，c = −5。代入公式：

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2)

x = (−3 ± √(9 + 40)) / 4 = (−3 ± √49) / 4

x = (−3 + 7) / 4 = 1 or x = (−3 − 7) / 4 = −5/2

So the solutions are x = 1 and x = −5/2.

所以解为 x = 1 和 x = −5/2。

6. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant. It tells us how many real roots a quadratic equation has.

根号内的表达式 b² − 4ac 称为判别式。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，方程有两个不同的实数根。
If b² − 4ac = 0, there is exactly one real root (a repeated root).
如果 b² − 4ac = 0，方程恰好有一个实数根（重根）。
If b² − 4ac < 0, there are no real roots.
如果 b² − 4ac < 0，方程没有实数根。

For example, the equation x² + 2x + 5 = 0 has discriminant 2² − 4 × 1 × 5 = 4 − 20 = −16, which is negative. So it has no real solutions.

例如，方程 x² + 2x + 5 = 0 的判别式为 2² − 4 × 1 × 5 = 4 − 20 = −16，是负数，因此没有实数解。

7. Solving Word Problems | 应用题求解

Quadratic equations frequently appear in word problems. You need to translate the given information into an equation, solve it, and then check whether the answers make sense in the context.

二次方程经常出现在应用题中。你需要将所给信息转化为方程，求解，然后检查答案在情境中是否合理。

Example: The area of a rectangle is 48 cm², and its length is 4 cm more than its width. Find the width.

例：一个矩形的面积是 48 cm²，长比宽多 4 cm。求宽。

Let the width be x cm. Then the length is x + 4 cm.

设宽为 x cm，则长为 x + 4 cm。

x(x + 4) = 48

x² + 4x − 48 = 0

Factorise: (x + 8)(x − 6) = 0.

因式分解：(x + 8)(x − 6) = 0。

So x = −8 or x = 6. Since width cannot be negative, the width is 6 cm.

所以 x = −8 或 x = 6。因为宽不能为负数，所以宽为 6 cm。

8. Drawing Quadratic Graphs | 二次函数图像

The graph of a quadratic function y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph.

二次函数 y = ax² + bx + c 的图像是抛物线。方程 ax² + bx + c = 0 的根就是图像与 x 轴的交点。

If a > 0, the parabola opens upwards. If a < 0, it opens downwards.

如果 a > 0，抛物线开口向上；如果 a < 0，抛物线开口向下。

To sketch the graph, you can find:

画草图时，你可以找到：

The roots (by solving the equation)
根（通过解方程）
The y-intercept (where x = 0)
y 轴截距（当 x = 0 时）
The vertex (by completing the square)
顶点（通过配方法）

For example, y = x² − 4x + 3 has roots x = 1 and x = 3, and its y-intercept is 3.

例如，y = x² − 4x + 3 的根为 x = 1 和 x = 3，y 轴截距为 3。

9. Common Mistakes | 常见错误

Many students lose marks on quadratic equations because of small but repeated errors. Here are the most common ones to avoid.

许多学生在二次方程上失分，是因为一些细小但反复出现的错误。以下是需要避免的常见错误。

Forgetting to rearrange into standard form before factorising.
因式分解前忘记整理成标准形式。
Sign errors when expanding brackets, especially with negative numbers.
展开括号时出现符号错误，尤其是负数时。
Forgetting to divide all terms by the coefficient of x² when using completing the square.
使用配方法时忘记除以 x² 的系数。
Writing x = 6 as the only answer when x = 0 is also possible.
当 x = 0 也可能是解时，只写出 x = 6 这一个答案。
Not checking answers by substituting back into the original equation.
没有将答案代入原方程进行检验。

10. Practice Questions | 练习题目

Try these questions on your own before checking the answers.

请先独立尝试这些题目，再核对答案。

1. Solve x² − 7x + 10 = 0.

1. 解方程 x² − 7x + 10 = 0。

2. Solve x² + 2x − 15 = 0.

2. 解方程 x² + 2x − 15 = 0。

3. Solve 3x² + 5x − 2 = 0 using the quadratic formula.

3. 用求根公式解 3x² + 5x − 2 = 0。

4. Find the value(s) of k such that the equation x² + kx + 9 = 0 has exactly one real root.

4. 求 k 的值，使得方程 x² + kx + 9 = 0 恰好有一个实数根。

Answers: 1. x = 2 or x = 5. 2. x = 3 or x = −5. 3. x = 1/3 or x = −2. 4. k = ±6.

答案：1. x = 2 或 x = 5。2. x = 3 或 x = −5。3. x = 1/3 或 x = −2。4. k = ±6。

11. Summary | 总结

Quadratic equations are an essential part of IGCSE Mathematics. You should be comfortable with the standard form, factorisation, completing the square, and the quadratic formula. The discriminant helps you predict the number of real roots, and word problems test your ability to apply these skills.

二次方程是 IGCSE 数学的重要组成部分。你应该熟练掌握标准形式、因式分解法、配方法和二次求根公式。判别式可以帮助你判断实数根的个数，而应用题则考查你运用这些技巧的能力。

Remember: always set the equation to zero before solving, check your answers, and think about whether the answers make sense in a real-world context.

请记住：求解前一定要将方程化为零的形式，检查你的答案，并思考答案在实际情境中是否合理。

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Solving Quadratic Equations | 二次方程求解