Solving Quadratic Equations | 二次方程的解法

📚 Solving Quadratic Equations | 二次方程的解法

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in almost every paper, whether in pure algebra, coordinate geometry, or problem-solving questions. Mastering the different methods of solving quadratics is essential for securing top marks.

二次方程是 IGCSE 数学中最重要的话题之一。无论是纯代数、坐标几何还是应用题，二次方程几乎出现在每份试卷中。掌握解二次方程的不同方法，是获得高分的关键。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. In its simplest form, it can be written as ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

二次方程是次数为 2 的多项式方程，即变量的最高次幂是 2。其最简形式可写成 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。若 a = 0，方程就变成一次方程，而不是二次方程。

For example, 2x² + 3x − 5 = 0 is quadratic, but 2x + 3 = 0 is linear. The value of the unknown, x, that satisfies the equation is called a root or solution of the equation.

例如，2x² + 3x − 5 = 0 是二次方程，而 2x + 3 = 0 是一次方程。满足方程的自变量 x 的值称为方程的根或解。

2. The Standard Form | 标准形式

Before solving a quadratic equation, it is often necessary to rearrange it into the standard form ax² + bx + c = 0. This means moving all terms to one side of the equals sign, with 0 on the other side, and collecting like terms.

在解二次方程之前，通常需要将其整理成标准形式 ax² + bx + c = 0。也就是说，把所有项移到等号的一侧，使另一侧为 0，并合并同类项。

x² − 3x = 4 → x² − 3x − 4 = 0

Here, a = 1, b = −3 and c = −4. Always check for a common factor first. For example, 2x² + 8x = 0 can be simplified by dividing both sides by 2 to give x² + 4x = 0, although it is not mandatory.

这里 a = 1，b = −3，c = −4。解题前应首先检查是否有公因数。例如，2x² + 8x = 0 可以两边同时除以 2，得到 x² + 4x = 0，当然这不是必须的。

3. Lesson 1: Solving by Factorisation | 方法一：因式分解法

Factorisation is the quickest method when the quadratic has simple integer coefficients. The idea is to rewrite the expression as a product of two brackets, then set each bracket equal to zero.

当二次方程具有简单的整数系数时，因式分解是最快捷的方法。其思路是将表达式改写为两个括号的乘积，然后令每个括号分别为零。

To factorise x² + bx + c, we look for two numbers that multiply to give c and add to give b. For example, solve x² + 5x + 6 = 0. We need two numbers whose product is 6 and whose sum is 5: these are 2 and 3.

要分解 x² + bx + c，我们需要找到两个数，它们的乘积等于 c，它们的和等于 b。例如，解 x² + 5x + 6 = 0。我们需要找两个数，其乘积为 6，其和为 5：这两个数是 2 和 3。

(x + 2)(x + 3) = 0

Now set each bracket to zero: x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3. Therefore, the solution set is x = −2 or x = −3.

现在令每个括号分别为零：x + 2 = 0 得 x = −2，x + 3 = 0 得 x = −3。因此解集为 x = −2 或 x = −3。

When the coefficient of x² is not 1, such as 2x² + 7x + 3 = 0, the factorisation requires more care. We look for factors of 2 × 3 = 6 that add to 7, which are 1 and 6, then split the middle term.

当 x² 的系数不是 1 时，例如 2x² + 7x + 3 = 0，因式分解需要更加小心。我们寻找 2 × 3 = 6 的因数，其和为 7，即 1 和 6，然后拆开中间项。

2x² + 7x + 3 = 2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

Thus (2x + 1)(x + 3) = 0, giving x = −½ or x = −3.

于是 (2x + 1)(x + 3) = 0，得到 x = −½ 或 x = −3。

4. Lesson 2: The Quadratic Formula | 方法二：公式法

Not all quadratic expressions can be factorised easily, especially when the roots are irrational or involve fractions. For any quadratic of the form ax² + bx + c = 0, the quadratic formula gives the solution directly.

并非所有二次多项式都能轻松因式分解，尤其是当根为无理数或涉及分数时。对于任何形式为 ax² + bx + c = 0 的二次方程，求根公式可以直接给出解。

x = (−b ± √(b² − 4ac)) / (2a)

This formula is given in the IGCSE formula sheet, but you must know how to substitute values correctly. Consider 2x² + 3x − 2 = 0. Here a = 2, b = 3 and c = −2. Substitute into the formula:

这个公式在 IGCSE 公式表中会提供，但你必须知道如何正确代入数值。考虑 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入公式：

x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4

So x = (−3 + 5)/4 = 2/4 = ½, or x = (−3 − 5)/4 = −8/4 = −2. Always simplify your answers where possible.

因此 x = (−3 + 5)/4 = 2/4 = ½，或 x = (−3 − 5)/4 = −8/4 = −2。记得尽可能化简答案。

5. Lesson 3: Completing the Square | 方法三：配方法

Completing the square is another algebraic technique that rewrites a quadratic expression in the form p(x + q)² + r. It is especially useful for finding turning points of parabolas and for solving equations that do not factorise.

配方法是另一种代数技巧，它将二次表达式改写为 p(x + q)² + r 的形式。它特别适用于求抛物线的顶点坐标，以及解无法因式分解的方程。

To complete the square for x² + bx + c, take half of b, square it, then adjust the constant. For example, solve x² + 6x + 8 = 0.

对于 x² + bx + c 配方，取 b 的一半，然后平方，再调整常数项。例如，解 x² + 6x + 8 = 0。

(x + 3)² − 9 + 8 = 0 → (x + 3)² − 1 = 0

Then (x + 3)² = 1, so x + 3 = ±1, giving x = −2 or x = −4. When a ≠ 1, factor out a first before completing the square inside the bracket.

然后 (x + 3)² = 1，所以 x + 3 = ±1，得 x = −2 或 x = −4。当 a ≠ 1 时，先把 a 提出来，再对括号内进行配方。

6. The Discriminant b² − 4ac | 判别式 b² − 4ac

Inside the quadratic formula, the expression b² − 4ac is called the discriminant. It tells us how many real roots a quadratic equation has, without fully solving it.

在求根公式中，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，而无需完整求解。

Discriminant 判别式	Nature of Roots 根的性质
b² − 4ac > 0	Two distinct real roots 两个不相等实数根
b² − 4ac = 0	One repeated real root 两个相等实数根（重根）
b² − 4ac < 0	No real roots 无实数根

For example, the equation x² + 4x + 5 = 0 has discriminant 4² − 4 × 1 × 5 = 16 − 20 = −4 < 0, so it has no real solutions.

例如，方程 x² + 4x + 5 = 0 的判别式为 4² − 4 × 1 × 5 = 16 − 20 = −4 < 0，因此该方程没有实数解。

7. Using the Discriminant in Problems | 判别式的应用

A common exam question gives a quadratic equation with an unknown parameter and asks for the value of that parameter such that the equation has a repeated root or no real roots.

考卷中常见的是给出含有未知参数的二次方程，然后要求求出该参数的值，使方程具有重根或无实数根。

Consider kx² + 6x + 3 = 0. If this equation has two equal roots, then the discriminant must equal zero. Here a = k, b = 6 and c = 3.

考虑 kx² + 6x + 3 = 0。若该方程有两个相等实根，则判别式必须为零。这里 a = k，b = 6，c = 3。

b² − 4ac = 0 → 6² − 4 × k × 3 = 0 → 36 − 12k = 0 → k = 3

So the value of k is 3. Always verify by substituting back into the original equation if time permits.

因此 k 的值为 3。如果时间允许，记得代回原方程进行验证。

8. Solving Word Problems with Quadratics | 二次方程应用题

Quadratic equations often arise from geometric or real-life problems. The key is to translate the words into an equation carefully, then solve it and check that the answer makes sense in context.

二次方程常从几何或实际生活问题中产生。关键在于仔细地把文字转化为方程，然后求解并检查答案是否符合实际情境。

For example, the area of a rectangle is 24 cm², and its length is 5 cm longer than its width. Let the width be x. Then the length is x + 5, so x(x + 5) = 24.

例如，一个矩形的面积为 24 cm²，其长比宽长 5 cm。设宽为 x，则长为 x + 5，因此 x(x + 5) = 24。

x² + 5x − 24 = 0 → (x + 8)(x − 3) = 0

Thus x = −8 or x = 3. Since a length cannot be negative, we reject x = −8 and conclude the width is 3 cm and the length is 8 cm. Always state your units and reject impossible answers.

因此 x = −8 或 x = 3。由于长度不能为负数，我们舍去 x = −8，得出宽为 3 cm，长为 8 cm。记得写明单位并舍去不合理的答案。

9. Quadratic Graphs and Roots | 二次函数图象与根

The solutions of a quadratic equation correspond to the x-intercepts of the graph of y = ax² + bx + c. The discriminant tells us how many times the curve crosses the x-axis.

二次方程的解对应着 y = ax² + bx + c 图象与 x 轴的交点横坐标。判别式告诉我们这条曲线与 x 轴相交多少次。

If b² − 4ac > 0, the parabola crosses the x-axis at two points.
若 b² − 4ac > 0，抛物线在两点处穿过 x 轴。
If b² − 4ac = 0, the parabola touches the x-axis at one point, which is the turning point.
若 b² − 4ac = 0，抛物线与 x 轴相切于一点，该点即为顶点。
If b² − 4ac < 0, the parabola does not touch the x-axis at all.
若 b² − 4ac < 0，抛物线与 x 轴完全没有交点。

This graphical interpretation helps you check whether your algebraic solutions are reasonable.

这种图象解释可以帮助你检查代数解是否合理。

10. Choosing the Right Method | 选择合适的方法

In the exam, you should choose the most efficient method for each question. Factorisation is fastest when the roots are integers or simple fractions. The quadratic formula always works, but it is slower. Completing the square is best when you also need the turning point.

在考试中，你应当针对每道题选择最有效的方法。当根为整数或简单分数时，因式分解最快。求根公式总是适用，但较慢。当还需要求顶点时，配方法是最佳选择。

Situation 情形	Recommended Method 推荐方法
Simple integer roots 简单整数根	Factorisation 因式分解
Ugly coefficients 系数复杂	Quadratic formula 求根公式
Need turning point 需要顶点	Completing the square 配方法

Whichever method you choose, always write down your working clearly. Marks are often awarded for method as well as the final answer.

无论选择哪种方法，务必清晰地写下你的解题步骤。评分时不仅看最终答案，也看重解题方法。

11. Common Mistakes and Final Tips | 常见错误与备考建议

Many students lose marks on quadratic equations because of small errors. The most frequent mistakes are forgetting to rearrange into standard form, dropping the ± sign in the quadratic formula, and dividing both sides of an equation by x, which loses a solution.

许多学生在二次方程上失分是因为小错误。最常见的错误包括：忘记整理成标准形式、在求根公式中漏掉 ± 号，以及将方程两边同时除以 x，导致丢失一个解。

Always bring all terms to one side before solving. 解方程前务必将所有项移到同一边。
Check whether x is a factor before dividing by it. 除以 x 之前先检查 x 是否为公因式。
Read the question: does it ask for exact values or decimal places? 仔细审题：要求精确值还是保留小数？
Verify your roots by substituting them back into the original equation. 将根代回原方程进行验证。

With regular practice, solving quadratic equations becomes a reliable source of marks. Make sure you can use all three methods fluently and interpret the discriminant correctly.

通过定期练习，解二次方程会成为稳定的得分点。请确保你能熟练使用三种方法，并正确理解判别式的含义。

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Solving Quadratic Equations | 二次方程的解法