Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, geometry, and everyday problem solving. Mastering how to solve them is essential for both Core and Extended papers, and exam questions often test multiple methods in a single problem.

二次方程是 IGCSE 数学中最重要的内容之一。它出现在代数、几何以及日常应用题中。掌握其解法对于 Core 和 Extended 两套试卷都至关重要，考试题目往往在一道题中综合考查多种方法。

1. The Standard Form | 标准形式

A quadratic equation is a polynomial equation of degree 2. Its general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The coefficient a in front of x² cannot be zero, otherwise the equation becomes linear. For example, 2x² − 5x + 3 = 0 has a = 2, b = −5 and c = 3.

二次方程是最高次数为 2 的多项式方程，一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。x² 前的系数 a 不能为零，否则方程就变成一次方程。例如 2x² − 5x + 3 = 0 中，a = 2，b = −5，c = 3。

2. Factorising when a = 1 | 当 a = 1 时的因式分解

When a = 1, we look for two numbers whose product is c and whose sum is b. For x² + 5x + 6 = 0, we need two numbers that multiply to 6 and add to 5. These are 2 and 3, so the equation becomes (x + 2)(x + 3) = 0. If the product of two factors is zero, one of them must be zero, so x = −2 or x = −3.

当 a = 1 时，我们要找两个数，使它们的乘积等于 c，和等于 b。对于 x² + 5x + 6 = 0，需要找两个相乘得 6、相加得 5 的数，那就是 2 和 3，于是方程化为 (x + 2)(x + 3) = 0。若两个因式的乘积为零，则必有一个因式为零，所以 x = −2 或 x = −3。

3. Factorising when a ≠ 1 | 当 a ≠ 1 时的因式分解

When the leading coefficient is not 1, we can use the grouping method. Take 2x² + 7x + 3 = 0. Multiply a and c: 2 × 3 = 6. Find two numbers that multiply to 6 and add to b = 7: they are 1 and 6. Rewrite the middle term: 2x² + x + 6x + 3 = 0, then group: x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0. Hence x = −1/2 or x = −3.

当首项系数不为 1 时，可以用分组分解法。以 2x² + 7x + 3 = 0 为例，先算 a × c = 2 × 3 = 6，再找两个相乘得 6、相加得 b = 7 的数，即 1 和 6。将中间项改写为 2x² + x + 6x + 3 = 0，然后分组：x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0。因此 x = −1/2 或 x = −3。

4. The Quadratic Formula | 二次公式

When factorising is difficult or impossible, the quadratic formula works for every quadratic equation. For ax² + bx + c = 0, the solutions are given by:

当因式分解困难或无法进行时，二次公式适用于一切二次方程。对于 ax² + bx + c = 0，其解为：

x = (−b ± √(b² − 4ac)) / 2a

To use it, substitute the values of a, b and c carefully. For example, solve 3x² − 5x − 2 = 0 with a = 3, b = −5, c = −2:

使用时，要小心代入 a、b、c 的值。例如解 3x² − 5x − 2 = 0，其中 a = 3，b = −5，c = −2：

x = (5 ± √(25 + 24)) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6

So x = 2 or x = −1/3. Always check that the equation is written in the form ax² + bx + c = 0 before substituting.

所以 x = 2 或 x = −1/3。代入之前，务必先把方程写成 ax² + bx + c = 0 的形式。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q. Starting from x² + bx, we add and subtract (b/2)²:

配方法把二次式改写为 (x + p)² + q 的形式。从 x² + bx 出发，我们加上并减去 (b/2)²：

x² + bx = (x + b/2)² − (b/2)²

For x² − 6x + 5 = 0, half of −6 is −3. So x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4. Setting this to zero gives (x − 3)² = 4, so x − 3 = ±2, meaning x = 5 or x = 1.

对于 x² − 6x + 5 = 0，−6 的一半是 −3。因此 x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4。令其等于零得 (x − 3)² = 4，故 x − 3 = ±2，即 x = 5 或 x = 1。

6. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant, often written as Δ. It tells us how many real roots exist without solving the equation:

二次公式中根号内的 b² − 4ac 称为判别式，通常记作 Δ。它无需解方程就能告诉我们方程有几个实根：

If Δ > 0, there are two distinct real roots. 若 Δ > 0，方程有两个不相等的实根。
If Δ = 0, there is exactly one repeated real root. 若 Δ = 0，方程有一个重根（两个相等的实根）。
If Δ < 0, there are no real roots. 若 Δ < 0，方程没有实根。

For example, 2x² + 4x + 2 = 0 has Δ = 16 − 16 = 0, so it has a repeated root at x = −1. The graph just touches the x-axis at this point.

例如，2x² + 4x + 2 = 0 的 Δ = 16 − 16 = 0，因此它在 x = −1 处有一个重根。图象在该点与 x 轴相切。

7. Solving by Graphing | 图象法

The roots of a quadratic equation are the x-coordinates where the curve y = ax² + bx + c crosses the x-axis. For y = x² − 4x + 3, the curve crosses the x-axis at x = 1 and x = 3, so x² − 4x + 3 = 0 has roots 1 and 3.

二次方程的根就是曲线 y = ax² + bx + c 与 x 轴交点的横坐标。对于 y = x² − 4x + 3，曲线在 x = 1 和 x = 3 处穿过 x 轴，因此 x² − 4x + 3 = 0 的根为 1 和 3。

The graph of a quadratic is a parabola. If a > 0, it opens upwards; if a < 0, it opens downwards. The axis of symmetry is x = −b/(2a), and the vertex lies on this line. In the IGCSE exam, you may be asked to read approximate roots from a given graph.

二次函数的图象是抛物线。若 a > 0，开口向上；若 a < 0，开口向下。对称轴为 x = −b/(2a)，顶点就在这条直线上。在 IGCSE 考试中，可能会要求你从给定图象中读出根的近似值。

8. Word Problems | 应用题

Quadratic equations often arise from geometric or real-life problems. Example: a rectangle has length 4 cm more than its width, and its area is 96 cm². Let the width be x. Then length is x + 4, and x(x + 4) = 96.

二次方程常由几何或实际问题产生。例：一个长方形的长比宽多 4 cm，面积为 96 cm²。设宽为 x，则长为 x + 4，于是 x(x + 4) = 96。

Expanding and rearranging: x² + 4x − 96 = 0. Factorising gives (x + 12)(x − 8) = 0, so x = 8 or x = −12. Since a length cannot be negative, the width is 8 cm and the length is 12 cm.

展开并整理得 x² + 4x − 96 = 0。因式分解得 (x + 12)(x − 8) = 0，所以 x = 8 或 x = −12。由于长度不能为负，宽为 8 cm，长为 12 cm。

In word problems, always discard any negative or non-physical solution and state the final answer with units.

在应用题中，务必舍弃负数或不符合实际的解，并在最终答案中写明单位。

9. Common Mistakes | 常见错误

Forgetting to rearrange the equation into ax² + bx + c = 0 before solving. 解题前忘记把方程整理成 ax² + bx + c = 0 的形式。
Sign errors when substituting negative values into the quadratic formula, for example writing −b as just b. 把负数代入二次公式时出现符号错误，例如把 −b 错写成 b。
Dropping the ± sign and losing one root. 漏写 ± 号，从而丢掉一个根。
Dividing both sides by x, which loses the root x = 0. 方程两边同除以 x，从而丢失 x = 0 这个根。

10. Exam Tips | 考试技巧

In the exam, show every line of working. If a question says “give your answers correct to 2 decimal places”, use the quadratic formula and round the final answers. When factorising, multiply your factors back to check. For graphs, label the axes and plot points accurately.

考试时，要写出每一步过程。若题目要求”答案精确到两位小数”，应使用二次公式并把最终答案四舍五入。因式分解后，把因式乘回去验证。作图时，标好坐标轴并准确描点。

A quick way to check your solutions is to substitute them back into the original equation. If both sides balance, your answer is correct. Also remember that a quadratic equation always has at most two solutions.

检验答案的快捷方法是把解代回原方程。若两边相等，则答案正确。还要记住，二次方程最多只有两个解。

11. Revision Summary | 复习总结

There are four main methods to solve a quadratic equation: factorising, the quadratic formula, completing the square and graphing. Factorising is fastest when simple factors exist; the quadratic formula works for all cases; completing the square is useful for finding vertices and solving inequalities; graphing gives a visual check.

解二次方程主要有四种方法：因式分解、二次公式、配方法和图象法。当存在简单因式时，因式分解最快；二次公式适用于所有情况；配方法常用于求顶点和解不等式；图象法能提供直观验证。

The discriminant Δ = b² − 4ac helps you quickly determine the nature of the roots. Practice applying each method to different examples until you can choose the most efficient approach automatically in the exam.

判别式 Δ = b² − 4ac 能帮助你快速判断根的性质。多练习用不同方法解各种例题，直到在考试中能自动选出最高效的方法。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations | 解一元二次方程