Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper. They test your ability to manipulate algebraic expressions, choose an appropriate solution method, and interpret results in context. Mastering them is essential for securing top grades.

二次方程几乎出现在每一份 IGCSE 数学试卷中。它考查你处理代数式、选择合适的解法以及在具体情境中解读结果的能力。掌握二次方程是冲击高分的关键。

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. The most general form is written as:

二次方程是次数为 2 的多项式方程，也就是说变量的最高次数是 2。它最一般的形式写作：

ax² + bx + c = 0

Here, a, b and c are constants, and the most important restriction is that a ≠ 0. If a were 0, the equation would become linear, not quadratic. The variable is usually x, but the same rules apply to any other letter such as t or y.

其中 a、b、c 为常数，最重要的限制条件是 a ≠ 0。如果 a 为 0，方程就变成一次方程而非二次方程。变量通常用 x 表示，但同样的规则也适用于 t 或 y 等其他字母。

For example, 2x² − 3x + 1 = 0 is quadratic, while 4x − 7 = 0 is linear because it has no x² term. Recognising the degree of an equation is the first step toward solving it correctly.

例如，2x² − 3x + 1 = 0 是二次方程，而 4x − 7 = 0 是一次方程，因为它没有 x² 项。正确判断方程的次数是求解的第一步。

2. The Standard Form and Coefficients | 标准形式与系数

Before solving a quadratic equation, you should 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 becomes 0, and collecting like terms.

在求解二次方程之前，你应该先把它整理成标准形式 ax² + bx + c = 0。这意味着把所有项移到等号的一侧，使另一侧变为 0，并合并同类项。

Consider the equation 3x² + 5x = 2 − 4x. To write it in standard form, add 4x to both sides and subtract 2 from both sides:

考虑方程 3x² + 5x = 2 − 4x。要把它写成标准形式，可以在两边同时加上 4x，并同时减去 2：

3x² + 9x − 2 = 0

Now you can read off the coefficients: a = 3, b = 9, c = −2. Pay close attention to the signs; a negative c often causes careless errors. In IGCSE exams, marks are frequently lost simply because a student forgot to rearrange the equation before applying a formula.

现在你可以读出系数：a = 3，b = 9，c = −2。要特别注意符号，负数 c 经常导致粗心错误。在 IGCSE 考试中，学生常常因为忘记先整理方程就直接套用公式而失分。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer factors. The idea is to write ax² + bx + c as a product of two brackets, then use the fact that if a product equals 0, at least one of the factors must be 0.

当二次式具有简单的整数因子时，因式分解是最快的方法。思路是把 ax² + bx + c 写成两个括号的乘积，然后利用“若乘积为 0，则至少有一个因子为 0”的性质。

For the equation x² + 5x + 6 = 0, we search for two numbers that multiply to give 6 and add to give 5. Those numbers are 2 and 3, so:

对于方程 x² + 5x + 6 = 0，我们需要找两个数，它们相乘得 6，相加得 5。这两个数就是 2 和 3，于是：

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

This gives two solutions:

由此得到两个解：

x + 2 = 0, so x = −2

x + 2 = 0，所以 x = −2
x + 3 = 0, so x = −3

x + 3 = 0，所以 x = −3

When the coefficient of x² is not 1, factorisation requires more care. For 2x² + 5x + 2 = 0, find two numbers that multiply to 2 × 2 = 4 and add to 5; those numbers are 4 and 1. Split the middle term:

当 x² 的系数不为 1 时，因式分解需要更加小心。对于 2x² + 5x + 2 = 0，我们需要找到两个数，相乘得 2 × 2 = 4，相加得 5，这两个数是 4 和 1。拆开中间项：

2x² + 4x + x + 2 = 0

Then factor by grouping: 2x(x + 2) + 1(x + 2) = 0, so (2x + 1)(x + 2) = 0. Hence x = −½ or x = −2.

然后分组提取公因式：2x(x + 2) + 1(x + 2) = 0，即 (2x + 1)(x + 2) = 0。因此 x = −½ 或 x = −2。

4. Solving by the Quadratic Formula | 公式法

The quadratic formula works for every quadratic equation, even when factorisation is impossible or very difficult. You do not need to memorise the derivation, but you must memorise the formula itself:

公式法适用于所有二次方程，即使因式分解行不通或非常困难时也能求解。你不需要记住推导过程，但必须牢记公式本身：

x = (−b ± √(b² − 4ac)) ÷ (2a)

To use the formula, substitute the values of a, b and c from the standard form. For the equation 2x² − 4x − 3 = 0, we have a = 2, b = −4, c = −3.

使用公式时，从标准形式中代入 a、b、c 的值。对于方程 2x² − 4x − 3 = 0，我们有 a = 2，b = −4，c = −3。

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

Simplifying √40 = 2√10, we obtain x = (4 ± 2√10) ÷ 4 = 1 ± √10 ÷ 2. As decimals, x ≈ 2.58 or x ≈ −0.58. In IGCSE, give exact surd answers unless the question asks for decimal approximations.

化简 √40 = 2√10，得到 x = (4 ± 2√10) ÷ 4 = 1 ± √10 ÷ 2。化成小数约为 x ≈ 2.58 或 x ≈ −0.58。在 IGCSE 中，除非题目要求近似值，否则应给出精确的根式答案。

5. Solving by Completing the Square | 配方法

Completing the square is another algebraic method that rewrites the quadratic in the form p(x + q)² + r. This method is particularly useful for finding turning points and is sometimes required explicitly by the exam question.

配方法是另一种代数方法，它把二次式改写成 p(x + q)² + r 的形式。这种方法在求顶点坐标时特别有用，有时考试题目也会明确要求使用配方法。

To complete the square for x² + 6x + 1 = 0, take half of 6, which is 3, and square it to get 9. Rewrite the equation as (x + 3)² − 9 + 1 = 0, giving (x + 3)² = 8.

对于 x² + 6x + 1 = 0 配方，取 6 的一半 3，平方得 9。把方程改写为 (x + 3)² − 9 + 1 = 0，即 (x + 3)² = 8。

Then take the square root of both sides, remembering the ± sign:

然后对两边开平方，记得加 ± 号：

x + 3 = ±√8 = ±2√2

Therefore x = −3 ± 2√2. This is the exact answer. Completing the square also allows you to read off the minimum point: for y = (x + 3)² − 8, the vertex is at (−3, −8).

因此 x = −3 ± 2√2。这就是精确答案。配方还能直接读出最小值点：对于 y = (x + 3)² − 8，顶点坐标为 (−3, −8)。

6. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant. It tells us about the nature of the roots without actually solving the equation. This is a favourite topic for multiple-choice and short-answer questions.

公式法根号内的表达式 b² − 4ac 称为判别式。不需要真正求解方程，它就能告诉我们根的性质。这是选择题和简答题的常见考点。

Discriminant	Nature of roots	Graph interpretation
b² − 4ac > 0	Two distinct real roots	The curve crosses the x-axis twice
b² − 4ac = 0	One repeated real root	The curve touches the x-axis once
b² − 4ac < 0	No real roots	The curve does not meet the x-axis

For example, the equation 3x² − 2x + 1 = 0 has discriminant (−2)² − 4 × 3 × 1 = 4 − 12 = −8, which is negative. Therefore the equation has no real roots, and its graph lies entirely above the x-axis.

例如，方程 3x² − 2x + 1 = 0 的判别式为 (−2)² − 4 × 3 × 1 = 4 − 12 = −8，是负数。因此该方程没有实数根，其图像完全位于 x 轴上方。

7. Applications: Word Problems | 应用题

Quadratic equations often arise in geometry and motion problems. A classic IGCSE question involves the area of a rectangle. For instance, a rectangle has length x + 4 cm, width x − 1 cm, and area 30 cm². Set up the equation:

二次方程经常出现在几何和运动问题中。一个经典的 IGCSE 题目涉及矩形的面积。例如，一个矩形的长为 x + 4 cm，宽为 x − 1 cm，面积为 30 cm²。列出方程：

(x + 4)(x − 1) = 30

Expanding gives x² + 3x − 4 = 30, so x² + 3x − 34 = 0. Factorising does not work here because 34 is prime, so use the quadratic formula with a = 1, b = 3, c = −34:

展开得 x² + 3x − 4 = 30，即 x² + 3x − 34 = 0。由于 34 是质数，因式分解行不通，因此使用公式法，其中 a = 1，b = 3，c = −34：

x = (−3 ± √145) ÷ 2

Since a length cannot be negative, we reject the negative root and take x = (−3 + √145) ÷ 2 ≈ 4.52. Always check whether your final answer makes sense in the context of the problem.

由于长度不能为负，我们舍去负根，取 x = (−3 + √145) ÷ 2 ≈ 4.52。始终要检查最终答案在题目情境中是否合理。

8. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards like a U; if a < 0, it opens downwards like an upside-down U. The solutions of the equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola.

函数 y = ax² + bx + c 的图像是抛物线。若 a > 0，抛物线开口朝上，形如 U；若 a < 0，开口朝下，形如倒 U。方程 ax² + bx + c = 0 的解对应抛物线与 x 轴的交点。

Key features you should be able to identify from the graph or equation:

你应该能够从图像或方程中识别以下关键特征：

The roots: where the curve crosses the x-axis

根：曲线与 x 轴的交点
The y-intercept: equal to c

y 轴截距：等于 c
The line of symmetry: x = −b ÷ (2a)

对称轴：x = −b ÷ (2a)
The vertex: minimum or maximum point of the curve

顶点：曲线的最高点或最低点

For y = x² − 4x + 3, the roots are x = 1 and x = 3, the y-intercept is 3, and the line of symmetry is x = 2. Substituting x = 2 gives y = 4 − 8 + 3 = −1, so the vertex is at (2, −1).

对于 y = x² − 4x + 3，根为 x = 1 和 x = 3，y 截距为 3，对称轴为 x = 2。代入 x = 2 得 y = 4 − 8 + 3 = −1，因此顶点坐标为 (2, −1)。

9. Common Mistakes to Avoid | 常见错误

Many students lose marks on quadratic equations due to a small number of repeated errors. Being aware of these traps before the exam can save you valuable points.

许多学生在二次方程上失分，往往是因为少数几个反复出现的错误。考前了解这些陷阱可以帮你保住宝贵的分数。

Forgetting the ± sign when taking a square root. For example, x² = 9 gives x = 3 AND x = −3.

开平方时忘记 ± 号。例如 x² = 9 的解应为 x = 3 和 x = −3 两个。
Using the quadratic formula before rearranging the equation into standard form. Always ensure one side equals 0 first.

未先把方程整理成标准形式就直接套用公式。务必先确保一边为 0。
Sign errors when substituting negative coefficients into the formula. Write out b, −b and 4ac separately.

代入负系数时出现符号错误。建议先把 b、−b 和 4ac 分别写清楚。
Concluding that a quadratic always has two distinct roots. Check the discriminant first.

误以为二次方程一定有两个不同的根。应先检查判别式。

10. Worked Examples and Exam Tips | 例题与考试技巧

Let us solve one exam-style problem using the most appropriate method. Solve 2x² − 5x = 3. First rearrange to standard form:

我们用最合适的方法解答一道考试风格的题目。解方程 2x² − 5x = 3。首先整理成标准形式：

2x² − 5x − 3 = 0

Factorisation is possible: look for two numbers multiplying to 2 × (−3) = −6 and adding to −5. Those are −6 and 1. Split the middle term:

这里可以因式分解：寻找两个数相乘得 2 × (−3) = −6，相加得 −5，这两个数是 −6 和 1。拆开中间项：

2x² − 6x + x − 3 = 0

Group the terms: 2x(x − 3) + 1(x − 3) = 0, so (2x + 1)(x − 3) = 0. Hence x = −½ or x = 3.

分组：2x(x − 3) + 1(x − 3) = 0，即 (2x + 1)(x − 3) = 0。因此 x = −½ 或 x = 3。

Exam tips to remember:

考试技巧总结：

Read the question carefully to see whether it asks for exact answers, surds, or decimal approximations.

仔细审题，看清题目要求的是精确值、根式还是小数近似值。
Choose the quickest reliable method: factorise if possible, otherwise use the formula, and use completing the square when the question demands it.

选择最快且可靠的方法：能因式分解就分解，否则用公式法，题目要求时才用配方法。
Show all working steps clearly. In IGCSE, method marks are awarded even if a final arithmetic slip occurs.

清楚展示每一步过程。在 IGCSE 中，即使最后有计算失误，步骤分仍然可以获得。
Check your answers by substituting them back into the original equation.

把答案代回原方程进行检验。

With regular practice, quadratic equations will become one of the most straightforward topics on your IGCSE paper. Master the methods, understand the discriminant, and always check your work.

通过定期练习，二次方程会成为你 IGCSE 试卷中最轻松的考点之一。掌握各种方法，理解判别式，并且永远记得检查你的答案。

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Solving Quadratic Equations | 解二次方程