Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

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 ≠ 0. Solving such equations is a core skill in IGCSE Mathematics, appearing in both Paper 1 and Paper 2 of the Edexcel syllabus.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。解这类方程是 IGCSE 数学的核心技能，在 Edexcel 考纲的 Paper 1 和 Paper 2 中都会出现。

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

Before solving, always rearrange the equation into the standard form ax² + bx + c = 0. The coefficient a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. For example, 2x² + 3x − 5 = 0 has a = 2, b = 3, c = −5.

在解题之前，务必先将方程整理为标准形式 ax² + bx + c = 0。系数 a 是二次项系数，b 是一次项系数，c 是常数项。例如 2x² + 3x − 5 = 0 中，a = 2，b = 3，c = −5。

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

如果 a = 0，方程就不是二次方程，而是一次方程。
The solutions of a quadratic equation are also called its roots, zeros, or x-intercepts.

二次方程的解也称为根、零点或与 x 轴的交点。

2. Solving by Factorisation | 因式分解法

Factorisation works when the quadratic expression can be written as a product of two linear factors. For example, solve x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5. These are 2 and 3, so x² + 5x + 6 = (x + 2)(x + 3) = 0.

因式分解法适用于二次表达式可以写成两个一次因式乘积的情况。例如解 x² + 5x + 6 = 0，我们需要找到两个数，它们相乘得 6，相加得 5。这两个数是 2 和 3，所以 x² + 5x + 6 = (x + 2)(x + 3) = 0。

Using the zero product property, if (x + 2)(x + 3) = 0, then either x + 2 = 0 or x + 3 = 0. Therefore x = −2 or x = −3.

根据零乘积性质，若 (x + 2)(x + 3) = 0，则要么 x + 2 = 0，要么 x + 3 = 0。因此 x = −2 或 x = −3。

Always check if the equation can be simplified first, e.g. divide by a common factor.

解题前先检查是否能化简，例如先除以公因数。
If the coefficient of x² is not 1, use the product-sum method or the ‘ac’ method.

如果 x² 的系数不为 1，可以使用乘积-和法或 ac 法。

Example: 2x² − 7x + 3 = 0 → (2x − 1)(x − 3) = 0 → x = ½ or x = 3

3. The Quadratic Formula | 二次公式法

When factorisation is difficult or impossible, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by:

当因式分解困难或无法进行时，使用二次公式。对于 ax² + bx + c = 0，解由以下公式给出：

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2. Substituting gives x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4, so x = ½ or x = −2.

例如解 2x² + 3x − 2 = 0，其中 a = 2，b = 3，c = −2。代入公式得 x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4，所以 x = ½ 或 x = −2。

Memorise the formula exactly – in Edexcel exams it is given on the formula sheet, but you must know how to use it.

准确记忆公式——Edexcel 考试中公式表会提供它，但你必须会使用。
Be careful with negative signs when substituting b and c.

代入 b 和 c 时小心负号。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form p(x + q)² + r. For x² + 6x + 5 = 0, we take half of 6, which is 3, and write (x + 3)² − 9 + 5 = 0, so (x + 3)² − 4 = 0.

配方法将二次式改写为 p(x + q)² + r 的形式。对于 x² + 6x + 5 = 0，取 6 的一半得 3，写成 (x + 3)² − 9 + 5 = 0，即 (x + 3)² − 4 = 0。

Then (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.

于是 (x + 3)² = 4，所以 x + 3 = ±2，得到 x = −1 或 x = −5。

This method is especially useful for finding the turning point of a quadratic graph.

这种方法在寻找二次函数图像的顶点时特别有用。
For ax² + bx + c, first factor out a from the first two terms.

对于 ax² + bx + c，先将 a 从前两项中提取出来。

5. The Discriminant | 判别式

The discriminant is the value D = b² − 4ac under the square root in the quadratic formula. It determines the nature of the roots:

判别式是二次公式中根号内的值 D = b² − 4ac，它决定了根的性质：

Discriminant	Nature of roots	判别式	根的性质
D > 0	Two real distinct roots	D > 0	两个不同实数根
D = 0	One real repeated root	D = 0	一个二重实数根
D < 0	No real roots	D < 0	无实数根

The discriminant also tells you whether the graph crosses the x-axis twice, touches it once, or never touches it.

判别式还能告诉你图像与 x 轴有两个交点、一个切点，还是没有交点。

6. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots are:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根的和与积为：

α + β = −b/a, αβ = c/a

These relationships can help you check your solutions or construct a quadratic equation when given the roots. For example, if the roots are 2 and −5, then sum = −3 and product = −10, so the equation is x² + 3x − 10 = 0 (multiply by a common factor if needed).

这些关系可以帮助你检验解，或在已知根时构造二次方程。例如，若根为 2 和 −5，则和 = −3，积 = −10，所以方程为 x² + 3x − 10 = 0（如需可用公因数调整）。

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

The roots of a quadratic equation are the x-coordinates where the graph of y = ax² + bx + c crosses the x-axis. The graph is a parabola. If a > 0 it opens upwards; if a < 0 it opens downwards.

二次方程的根就是二次函数 y = ax² + bx + c 的图像与 x 轴交点的横坐标。图像是抛物线。若 a > 0 开口向上；若 a < 0 开口向下。

The turning point (vertex) is at x = −b/(2a).

顶点（驻点）的 x 坐标为 x = −b/(2a)。
The y-intercept is always c.

y 轴截距始终为 c。
If the equation has no real roots, the graph does not cross the x-axis; it lies entirely above (a > 0) or below (a < 0) the axis.

若方程无实数根，图像不与 x 轴相交；它完全位于 x 轴上方（a > 0）或下方（a < 0）。

Vertex form: y = a(x − h)² + k, where (h, k) is the vertex.

8. Solving Word Problems | 应用题求解

Many IGCSE questions present quadratic equations in context, such as area, projectile motion, or consecutive integers. The key steps are:

许多 IGCSE 题目会以实际背景给出二次方程，例如面积、抛体运动或连续整数。关键步骤如下：

Define the variable clearly.

明确定义变量。
Translate the information into a quadratic equation.

将已知信息转化为二次方程。
Solve the equation using an appropriate method.

使用合适的方法解方程。
Check whether both solutions make sense in the context (e.g. length cannot be negative).

检查两个解在情境中是否合理（例如长度不能为负）。

Example: The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Let width = x, then length = x + 3. Hence x(x + 3) = 40, so x² + 3x − 40 = 0. Factorising gives (x + 8)(x − 5) = 0, so x = 5 (reject x = −8). Width = 5 cm, length = 8 cm.

例：一个长方形的长比宽多 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3。因此 x(x + 3) = 40，即 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = 5（舍去 x = −8）。宽为 5 cm，长为 8 cm。

9. Common Mistakes and Tips | 常见错误与建议

Students often lose marks on quadratic equations due to avoidable errors. Here are the most common ones:

学生在二次方程题目中常因可避免的错误失分。以下是最常见的错误：

Forgetting to rearrange into standard form before factorising or using the formula.

在因式分解或使用公式前忘记整理为标准形式。
Missing the ± sign when taking square roots.

开平方时遗漏 ± 号。
Incorrectly simplifying fractions in the quadratic formula.

在二次公式中错误化简分数。
Not checking whether a root is invalid in a word problem.

在应用题中未检查根是否无效。
Mixing up the signs when using the sum and product of roots.

使用根的和与积时搞错符号。

Always double-check your answers by substituting back into the original equation.

始终将解代回原方程进行检验。

10. Worked Exam-Style Questions | 典型考题演练

Let us solve a complete Edexcel-style problem. Solve 3x² − 5x − 2 = 0 by factorisation.

让我们完成一道 Edexcel 风格的典型题目。用因式分解法解 3x² − 5x − 2 = 0。

We need two numbers whose product is 3 × (−2) = −6 and whose sum is −5. The numbers are −6 and 1. Rewrite the middle term: 3x² − 6x + x − 2 = 0. Factorise by grouping: 3x(x − 2) + 1(x − 2) = 0, so (3x + 1)(x − 2) = 0. Thus x = −1/3 or x = 2.

我们需要两个数，其积为 3 × (−2) = −6，其和为 −5。这两个数是 −6 和 1。将中间项改写为 3x² − 6x + x − 2 = 0。分组因式分解：3x(x − 2) + 1(x − 2) = 0，所以 (3x + 1)(x − 2) = 0。因此 x = −1/3 或 x = 2。

Second example (non-calculator): Solve x² + 4x + 1 = 0, giving your answers to 2 decimal places. Use the formula with a = 1, b = 4, c = 1:

第二个例子（非计算器）：解 x² + 4x + 1 = 0，答案保留两位小数。使用公式，a = 1，b = 4，c = 1：

x = (−4 ± √(16 − 4)) / 2 = (−4 ± √12) / 2 = (−4 ± 2√3) / 2 = −2 ± √3

Therefore x ≈ −2 + 1.732 = −0.27 or x ≈ −2 − 1.732 = −3.73.

因此 x ≈ −2 + 1.732 = −0.27 或 x ≈ −2 − 1.732 = −3.73。

11. Strategy Summary | 解题策略总结

Choose the most efficient method based on the equation:

根据方程选择最高效的方法：

Method	When to use	方法	使用时机
Factorisation	When the quadratic factorises easily	因式分解	二次式容易分解时
Quadratic formula	When factorisation is not obvious	二次公式	因式分解不明显时
Completing the square	When finding the vertex or solving with surds	配方法	求顶点或含根式解时

Remember that in the Edexcel IGCSE calculator paper, you can use your calculator’s equation solver to check your answer, but you must still show your working.

记住，在 Edexcel IGCSE 允许使用计算器的试卷中，你可以使用计算器的方程求解功能检验答案，但仍需写出解题过程。

12. Final Revision Checklist | 考前自检清单

Before the exam, make sure you can:

考试前，确保你能做到：

Rearrange any quadratic equation into standard form.

将任意二次方程整理为标准形式。
Solve by factorisation, formula, and completing the square.

用因式分解、公式法和配方法求解。
Interpret the discriminant correctly.

正确解释判别式的含义。
Use sum and product of roots in simple problems.

在简单问题中使用根的和与积。
Connect roots to the graph of y = ax² + bx + c.

将根与 y = ax² + bx + c 的图像联系起来。
Solve realistic word problems and reject invalid solutions.

解决实际应用题并舍去无效解。

With consistent practice, quadratic equations will become one of your most reliable topics in the IGCSE Mathematics exam.

通过持续练习，二次方程将成为你在 IGCSE 数学考试中最有把握的板块之一。

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Solving Quadratic Equations | 解二次方程