Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in the Edexcel IGCSE Mathematics syllabus. Mastering them builds a strong foundation for algebra, graphs, and problem-solving. This guide explains every key method you need, with clear examples and examiner-style tips.

二次方程是 Edexcel IGCSE 数学课程中最重要的主题之一。掌握它能为代数、图像和问题解决打下坚实基础。本指南讲解你所需的一切关键方法，并配有清晰的例题和考官风格的技巧。

1. The Standard Form | 标准形式

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants, and a is not equal to 0. The highest power of x is 2, which gives the equation its name.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。未知数 x 的最高次数是 2，所以称为“二次”方程。

ax² + bx + c = 0 (a ≠ 0)

In IGCSE questions, you must first rearrange the equation so that one side is 0. For example, x² + 3x = 4 becomes x² + 3x − 4 = 0.

在 IGCSE 题目中，你必须先将方程整理成一边为 0 的形式。例如，x² + 3x = 4 应写成 x² + 3x − 4 = 0。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic has simple integer roots. The idea is to express ax² + bx + c as a product of two binomials.

当二次方程有简单的整数根时，因式分解往往是最快的方法。思路是将 ax² + bx + c 表示为两个二项式的乘积。

For example, to solve x² + 5x + 6 = 0, find two numbers that multiply to 6 and add to 5. These are 2 and 3, so the equation becomes (x + 2)(x + 3) = 0.

例如，解 x² + 5x + 6 = 0 时，找到两个数相乘得 6、相加得 5。这两个数是 2 和 3，因此方程变为 (x + 2)(x + 3) = 0。

Then apply the zero-product property: if a product is 0, one of the factors must be 0. Hence x = −2 or x = −3.

然后应用零乘积性质：如果乘积为 0，则至少有一个因数为 0。因此 x = −2 或 x = −3。

Steps for factorising:

因式分解步骤：

Arrange the equation as ax² + bx + c = 0. 将方程整理为 ax² + bx + c = 0。
For the simple case a = 1, find two integers whose product is c and sum is b. 对于 a = 1 的简单情形，找出两个整数，使它们的乘积为 c、和为 b。
Write the factors and set each equal to 0. 写出因式并令每个因式等于 0。
Solve the resulting linear equations. 解出所得的一次方程。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful for finding turning points and for solving equations when factorisation is difficult.

配方法将二次式改写为 a(x + p)² + q 的形式。这有助于求顶点，以及在因式分解困难时解方程。

Start with x² + 6x − 1 = 0. Take half of the coefficient of x, which is 3, and write (x + 3)². Expanding gives x² + 6x + 9, which is 10 more than the original x² + 6x − 1. So subtract 10:

从 x² + 6x − 1 = 0 开始。取 x 系数的半，即 3，写成 (x + 3)²。展开得 x² + 6x + 9，比原来的 x² + 6x − 1 多 10，因此减去 10：

(x + 3)² − 10 = 0

Then (x + 3)² = 10, so x + 3 = ±√10, giving x = −3 ± √10.

于是 (x + 3)² = 10，所以 x + 3 = ±√10，即 x = −3 ± √10。

4. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation, even when factorisation is impossible. It is derived by completing the square on the general form.

二次公式适用于所有二次方程，即使无法因式分解。它是通过对一般形式进行配方推导得到的。

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, c = −3. Substitute into the formula:

例如，解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入公式：

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

Simplifying gives x = (2 ± √10) / 2. Always check whether the question asks for exact values or decimal approximations.

化简得 x = (2 ± √10) / 2。始终注意题目要求的是精确值还是小数近似值。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: Δ = b² − 4ac. It tells us the nature of the roots without solving the equation.

判别式是二次公式中根号内的部分：Δ = b² − 4ac。它无需解方程就能告诉我们根的性质。

Discriminant 判别式	Nature of roots 根的性质
Δ > 0	Two distinct real roots 两个不同实根
Δ = 0	One repeated real root 一个重根
Δ < 0	No real roots 没有实根

If Δ is a perfect square, the roots are rational and factorisation is possible. If Δ is positive but not a perfect square, the roots are irrational.

如果 Δ 是完全平方数，根是有理数，可以因式分解；如果 Δ 为正但不是完全平方数，根是无理数。

6. Roots of Quadratic Equations | 方程的根

For a quadratic equation ax² + bx + c = 0, the sum of the roots is −b/a and the product of the roots is c/a. This relationship comes from expanding a(x − α)(x − β).

对于二次方程 ax² + bx + c = 0，两根之和为 −b/a，两根之积为 c/a。这一关系来自展开 a(x − α)(x − β)。

These results help you check solutions or construct a quadratic equation when the roots are known.

这些结果有助于检验解，或在已知根时构造二次方程。

Sum of roots 根之和: α + β = −b/a

Product of roots 根之积: αβ = c/a

7. Quadratic Graphs and Roots | 二次函数图像与根

The graph of y = ax² + bx + c is a parabola. The x-coordinates of the points where the graph crosses the x-axis are exactly the roots of the equation ax² + bx + c = 0.

y = ax² + bx + c 的图像是抛物线。图像与 x 轴交点的横坐标正好是方程 ax² + bx + c = 0 的根。

If the discriminant is negative, the graph does not cross the x-axis. If the discriminant is zero, the graph touches the x-axis at one point, which is the turning point.

如果判别式为负，图像不与 x 轴相交；如果判别式为零，图像与 x 轴相切于一点，该点就是顶点。

The line of symmetry is x = −b/(2a). This is also the x-coordinate of the vertex.

对称轴为 x = −b/(2a)，这也是顶点的 x 坐标。

8. Word Problems | 文字题

Quadratic equations often arise from geometric or real-life situations. You need to translate the words into an equation, solve it, and then interpret the answers in context.

二次方程常常来自几何或现实情境。你需要将文字翻译成方程，求解，然后结合背景解释答案。

Example: The length of a rectangle is 3 cm longer than its width, and the area is 40 cm². Let the width be x. Then the length is x + 3, so x(x + 3) = 40.

例如：一个长方形的长比宽长 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3，所以 x(x + 3) = 40。

Rearranging: x² + 3x − 40 = 0, which factorises to (x + 8)(x − 5) = 0. The solutions are x = −8 and x = 5. Since a length cannot be negative, the width is 5 cm and the length is 8 cm.

整理得 x² + 3x − 40 = 0，因式分解为 (x + 8)(x − 5) = 0。解为 x = −8 和 x = 5。由于长度不能为负，所以宽为 5 cm，长为 8 cm。

9. Common Mistakes | 常见错误

Many students lose marks in Edexcel IGCSE because of avoidable errors. Here are the most common pitfalls and how to avoid them.

许多学生在 Edexcel IGCSE 中因可避免的错误而失分。以下是最常见的陷阱及避免方法。

Forgetting to rearrange the equation to 0 before factorising. 因式分解前忘记将方程整理为 0。
Missing the negative root: when x² = k, x = ±√k. 漏掉负根：当 x² = k 时，x = ±√k。
Incorrectly using the quadratic formula when a, b or c have negative signs. 当 a、b、c 为负数时错误使用二次公式。
Dropping the ± sign when taking square roots. 取平方根时丢掉 ± 号。
Ignoring the physical meaning of a negative answer in word problems. 在文字题中忽略负答案的实际意义。

Always substitute your answers back into the original equation to verify them.

始终将答案代回原方程进行验证。

10. Practice Points | 练习要点

To succeed with quadratic equations, practice each method until it becomes automatic. In the exam, choose the fastest method that fits the question.

要学好二次方程，请将每种方法练习到熟练为止。考试时，选择最适合题目且最快的方法。

Quick decision guide:

快速选择指南：

If the equation factorises easily, use factorisation. 如果方程容易因式分解，就用因式分解法。
If you need the turning point, complete the square. 如果需要顶点，使用配方法。
If you are unsure or the coefficients are messy, use the quadratic formula. 如果不确定或系数较复杂，使用二次公式。

Remember that not every question requires a calculator. Practise exact arithmetic with surds as well.

记住，不是每题都允许使用计算器。也要练习包含根号的精确运算。

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Solving Quadratic Equations | 解二次方程