Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear frequently in IGCSE Mathematics. Mastering the methods of solving them is essential for success in both non-calculator and calculator papers. This article reviews the standard techniques, the discriminant, and real-world applications.

二次方程在 IGCSE 数学中频繁出现。掌握其解法对于在非计算器和计算器试卷中取得好成绩都至关重要。本文复习标准解法、判别式以及实际应用。

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

A quadratic equation is a polynomial equation of degree 2, generally written in the form ax² + bx + c = 0, where a, b, c are constants and a ≠ 0. The term “quadratic” comes from “quadratus”, meaning square, because the highest power of the unknown is 2.

二次方程是次数为 2 的多项式方程，通常写成 ax² + bx + c = 0 的形式，其中 a、b、c 为常数且 a ≠ 0。术语“二次”源自拉丁语“quadratus”，意为“平方”，因为未知数的最高次数是 2。

For example, 2x² – 3x + 1 = 0 is quadratic, while x³ – 4x = 0 is cubic. Some quadratics may appear in factored form such as (x – 2)(x + 5) = 0.

例如，2x² – 3x + 1 = 0 是二次方程，而 x³ – 4x = 0 是三次方程。有些二次方程可能以因式形式出现，如 (x – 2)(x + 5) = 0。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic expression can be written as a product of two linear factors. The key principle is the zero-product property: if AB = 0, then A = 0 or B = 0.

当二次表达式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。关键原理是零乘积性质：若 AB = 0，则 A = 0 或 B = 0。

Example: Solve x² – 5x + 6 = 0.

例：解方程 x² – 5x + 6 = 0。

Find two numbers whose product is 6 and sum is -5. They are -2 and -3. Hence:

找两个数，其乘积为 6，和为 -5。它们是 -2 和 -3。因此：

x² – 5x + 6 = (x – 2)(x – 3) = 0

Thus x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

因此 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

When the coefficient of x² is not 1, we may need to factor by grouping or use trial and error. For example, 2x² + 5x + 2 = (2x + 1)(x + 2).

当 x² 的系数不为 1 时，可能需要用分组分解法或尝试法。例如，2x² + 5x + 2 = (2x + 1)(x + 2)。

Always expand your factors to check the answer.
If the equation is not in standard form, rearrange it first.
Some quadratics cannot be factored over integers; then use another method.

始终展开因式以检查答案。
若方程不是标准形式，先整理。
有些二次式无法用整数因式分解，此时需用其他方法。

3. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is especially useful for finding turning points and solving equations that do not factorise neatly.

配方法将二次式改写为 a(x + p)² + q 的形式。这在求顶点坐标以及解不能整洁分解的方程时尤其有用。

Example: Solve x² + 6x – 7 = 0 by completing the square.

例：用配方法解 x² + 6x – 7 = 0。

First, move the constant term to the other side:

首先，将常数项移到另一边：

x² + 6x = 7

Add (6/2)² = 9 to both sides:

两边同时加上 (6/2)² = 9：

x² + 6x + 9 = 16

Then (x + 3)² = 16, so x + 3 = ±4. Hence x = 1 or x = -7.

于是 (x + 3)² = 16，所以 x + 3 = ±4。因此 x = 1 或 x = -7。

For a general quadratic ax² + bx + c = 0, the completed square form is:

对于一般二次方程 ax² + bx + c = 0，配方形式为：

a[(x + b/(2a))² – (b² – 4ac)/(4a²)] = 0

This method also reveals the coordinates of the vertex: (-b/(2a), c – b²/(4a)) after factoring out a.

此方法还可揭示顶点坐标：提取 a 后为 (-b/(2a), c – b²/(4a))。

4. The Quadratic Formula | 二次公式

The quadratic formula is a universal method that works for any quadratic equation. It is derived by completing the square of ax² + bx + c = 0.

二次公式是一种通用方法，适用于任何二次方程。它通过对 ax² + bx + c = 0 配方推导得出。

x = (-b ± √(b² – 4ac)) / (2a)

Here the symbol ± means that there are usually two solutions: one with plus, one with minus.

这里 ± 表示通常有两个解：一个取加号，一个取减号。

Example: Solve 2x² – 4x – 3 = 0 using the formula.

例：用公式解 2x² – 4x – 3 = 0。

Here a = 2, b = -4, c = -3. Substitute:

这里 a = 2，b = -4，c = -3。代入：

x = (4 ± √((-4)² – 4·2·(-3))) / (2·2)

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

Since √40 = 2√10, we have x = (4 ± 2√10)/4 = 1 ± √10/2. Approximate answers: x ≈ 2.58 or x ≈ -0.58.

因为 √40 = 2√10，我们有 x = (4 ± 2√10)/4 = 1 ± √10/2。近似解：x ≈ 2.58 或 x ≈ -0.58。

Check whether the question asks for exact values or decimals.
Always write a, b, c clearly before substituting to avoid sign errors.
Use the formula when factorisation is difficult or impossible.

注意题目要求精确值还是小数。
代入前明确写出 a、b、c，避免符号错误。
当分解困难或不可能时，使用公式。

5. The Discriminant | 判别式

The discriminant is the expression b² – 4ac inside the quadratic formula. Its value tells us the nature of the roots without fully solving the equation.

判别式是二次公式中根号内的表达式 b² – 4ac。其值无需完整解方程即可告诉我们根的性质。

Discriminant Δ	Nature of roots	Graphical meaning
Δ > 0	Two distinct real roots	Parabola crosses x-axis twice
Δ = 0	One repeated real root	Parabola touches x-axis once (tangent)
Δ < 0	No real roots (two complex roots)	Parabola does not intersect x-axis

判别式 Δ	根的性质	图像意义
Δ > 0	两个不等实根	抛物线与 x 轴有两个交点
Δ = 0	一个重根（两个相等实根）	抛物线与 x 轴相切于一点
Δ < 0	没有实根（两个复数根）	抛物线与 x 轴无交点

For example, the equation x² + x + 1 = 0 has Δ = 1 – 4 = -3, so it has no real roots. The equation x² – 4x + 4 = 0 has Δ = 0, so it has one repeated root x = 2.

例如，方程 x² + x + 1 = 0 的 Δ = 1 – 4 = -3，因此没有实根。方程 x² – 4x + 4 = 0 的 Δ = 0，因此有一个重根 x = 2。

6. Graphs and Roots | 图像与根

The solutions of a quadratic equation ax² + bx + c = 0 correspond to the x-intercepts of the parabola y = ax² + bx + c. The graph is symmetric about the vertical line x = -b/(2a), which is the axis of symmetry.

二次方程 ax² + bx + c = 0 的解对应于抛物线 y = ax² + bx + c 的 x 轴交点。图像关于垂直线 x = -b/(2a) 对称，该线称为对称轴。

If a > 0, the parabola opens upwards and has a minimum point; if a < 0, it opens downwards and has a maximum point. The y-coordinate of the vertex is the value of the expression at x = -b/(2a).

若 a > 0，抛物线开口向上，有最小值点；若 a < 0，开口向下，有最大值点。顶点的 y 坐标是表达式在 x = -b/(2a) 处的值。

When the discriminant is positive, the parabolic curve crosses the x-axis twice. When it is zero, the vertex lies on the x-axis. When negative, the graph stays entirely above or below the x-axis depending on the sign of a.

当判别式为正时，抛物线与 x 轴相交两次。当为零时，顶点位于 x 轴上。当为负时，图像完全位于 x 轴上方或下方，具体取决于 a 的符号。

Sketching a quadratic graph typically involves finding the roots, the y-intercept (c), and the vertex. This helps in solving inequalities such as ax² + bx + c > 0.

绘制二次函数图像通常需要找到根、y 轴截距 (c) 以及顶点。这也有助于解不等式，如 ax² + bx + c > 0。

7. Word Problems and Applications | 应用题与实际应用

Many real-world problems involve quadratic equations. For instance, the height h(t) of a projectile thrown upward can be modelled by h(t) = -gt² + v₀t + h₀, where g is gravity, v₀ is initial velocity, and h₀ is initial height.

许多实际问题涉及二次方程。例如，抛射物的高度 h(t) 可由 h(t) = -gt² + v₀t + h₀ 建模，其中 g 为重力加速度，v₀ 为初速度，h₀ 为初始高度。

In geometry, the area of a rectangle with length and width expressed in terms of x may lead to a quadratic equation. For example, a rectangle’s length is 3 cm more than its width, and its area is 40 cm². Let width = x, then x(x + 3) = 40, so x² + 3x – 40 = 0.

在几何中，长度和宽度用 x 表示的矩形面积可能导出二次方程。例如，一个矩形的长比宽多 3 cm，面积为 40 cm²。设宽为 x，则 x(x + 3) = 40，即 x² + 3x – 40 = 0。

Factorising gives (x + 8)(x – 5) = 0, so x = 5 (rejecting the negative value because width cannot be negative). Thus the width is 5 cm and the length is 8 cm.

因式分解得 (x + 8)(x – 5) = 0，所以 x = 5（舍去负值，因为宽不能为负）。因此宽为 5 cm，长为 8 cm。

When solving word problems, always check whether both mathematical solutions are practical. Often the negative root is discarded.

解决应用题时，务必检查两个数学解是否都符合实际。通常负根会被舍弃。

8. Common Mistakes and Tips | 常见错误与技巧

Students often make errors when the coefficient a is negative, when they forget to set the equation to zero before factorising, or when they mishandle the square root in the quadratic formula.

学生常犯的错误包括：a 为负数时处理不当、分解前忘记将方程整理为零、以及在二次公式中错误处理根号。

Mistake: Dividing both sides by x when x is a variable. This loses the root x = 0.
Correct: Rearrange and factorise: x² – 3x = 0 gives x(x – 3) = 0, so x = 0 or x = 3.

错误：当 x 是变量时两边同除以 x。这会导致丢失根 x = 0。
正确：整理并分解：x² – 3x = 0 得 x(x – 3) = 0，所以 x = 0 或 x = 3。

Mistake: Using the quadratic formula with the wrong signs, e.g. writing -b instead of (-b).
Tip: Always substitute carefully and use brackets around negative values.

错误：使用二次公式时符号出错，例如把 -b 当成 (-b)。
技巧：仔细代入，并在负值外用括号。

Mistake: Ignoring the plus/minus symbol when extracting square roots.
Correct: From (x + 2)² = 9, write x + 2 = ±3, then solve both equations.

错误：开平方时忽略正负号。
正确：由 (x + 2)² = 9，写出 x + 2 = ±3，然后分别解两个方程。

In the non-calculator paper, practise mental arithmetic of squares and simple factors. In the calculator paper, know how to use your calculator’s equation solver, but always show your working to earn full marks.

在非计算器试卷中，练习平方数和简单因式的口算。在计算器试卷中，知道如何使用计算器的方程求解功能，但始终写出过程以获得满分。

9. Practice Questions | 练习题目

Try these questions to test your understanding. Solutions are given below.

尝试以下题目以检验你的理解。答案附后。

1. Solve by factorisation: x² – 7x + 10 = 0

1. 用因式分解法解：x² – 7x + 10 = 0

2. Solve by completing the square: x² – 4x – 5 = 0

2. 用配方法解：x² – 4x – 5 = 0

3. Use the quadratic formula to solve 3x² + 2x – 8 = 0.

3. 用二次公式解 3x² + 2x – 8 = 0。

4. Find the value of k for which the equation kx² + 3x + 1 = 0 has two equal roots.

4. 求 k 的值，使得方程 kx² + 3x + 1 = 0 有两个相等实根。

Solutions:

答案：

1. x = 2 or x = 5. 2. x = 5 or x = -1. 3. x = 4/3 or x = -2. 4. Discriminant = 9 – 4k = 0, so k = 9/4.

1. x = 2 或 x = 5。2. x = 5 或 x = -1。3. x = 4/3 或 x = -2。4. 判别式 = 9 – 4k = 0，所以 k = 9/4。

10. Final Revision Summary | 最后复习总结

To solve a quadratic equation, first check if factorisation works quickly. If not, use completing the square or the quadratic formula. Always interpret the discriminant to understand the nature of roots, and remember to reject solutions that do not make sense in context.

解二次方程时，首先检查是否能快速分解。若不能，则使用配方法或二次公式。始终通过判别式理解根的性质，并记住舍去在情境中无意义的解。

Practice with a variety of questions, including those with coefficients that are not integers, and those that require rearranging. Familiarity with graphs helps visualise results and avoid sign errors.

练习各类题目，包括系数不是整数以及需要先整理的题目。熟悉图像有助于直观理解结果并避免符号错误。

For IGCSE exams, set out your working clearly and use the method specified in the question if there is one. Good luck!

对于 IGCSE 考试，请清晰列出计算步骤，如果题目指定方法则按指定方法作答。祝你好运！

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Solving Quadratic Equations | 解一元二次方程