Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are a fundamental topic in IGCSE Mathematics. They appear in algebra, geometry, and even in real-world problem solving. This article provides a comprehensive guide to understanding, solving, and applying quadratic equations, aligned with the IGCSE syllabus.

二次方程是IGCSE数学中的一个基础主题。它们出现在代数、几何甚至现实世界的问题解决中。本文提供了一份关于理解、求解和应用二次方程的全面指南，与IGCSE教学大纲保持一致。

1. Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is any equation that can be written in the standard form:

二次方程是任何可以写成标准形式的方程：

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.

其中 a、b 和 c 是常数，且 a ≠ 0。 ax² 是二次项，bx 是一次项，c 是常数项。

For example, 2x² + 5x – 3 = 0 is a quadratic equation. If a = 0, the equation becomes linear, so the condition a ≠ 0 is essential.

例如，2x² + 5x – 3 = 0 是一个二次方程。如果 a = 0，方程就变成了线性方程，因此条件 a ≠ 0 是必不可少的。

2. Factorization Method | 因式分解法

Factoring is often the quickest way to solve a quadratic equation when the factors are easy to identify. The idea is to rewrite the quadratic expression as a product of two linear factors.

因式分解通常是解二次方程最快的方法，特别是在因子容易识别的情况下。其思想是将二次表达式重写为两个线性因子的乘积。

For a quadratic equation in the form x² + bx + c = 0 (where a = 1), find two numbers that multiply to give c and add to give b.

对于形式为 x² + bx + c = 0 的二次方程（其中 a = 1），找到两个数，它们的乘积等于 c，且它们的和等于 b。

Example: Solve x² – 5x + 6 = 0.

示例：解 x² – 5x + 6 = 0。

We need two numbers whose product is 6 and sum is -5. These numbers are -2 and -3. Thus:

我们需要找到两个数，乘积为 6，和为 -5。这两个数是 -2 和 -3。因此：

(x – 2)(x – 3) = 0

Using the zero product rule, either x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.

根据零乘积规则，要么 x – 2 = 0，要么 x – 3 = 0，所以 x = 2 或 x = 3。

When a ≠ 1, the method is slightly more involved. For example, solve 2x² + 7x + 3 = 0. We look for factors of 2×3 = 6 that add to 7. The numbers are 1 and 6. Then split the middle term:

当 a ≠ 1 时，方法稍微复杂一些。例如，解 2x² + 7x + 3 = 0。我们寻找 2×3 = 6 的因子且和为 7。这两个数是 1 和 6。然后拆分中项：

2x² + x + 6x + 3 = 0

Now factor by grouping: x(2x + 1) + 3(2x + 1) = 0, so (x + 3)(2x + 1) = 0. Hence x = -3 or x = -½.

现在分组因式分解：x(2x + 1) + 3(2x + 1) = 0，所以 (x + 3)(2x + 1) = 0。因此 x = -3 或 x = -½。

3. Completing the Square | 配方法

Completing the square is a powerful technique that transforms a quadratic equation into a perfect square trinomial plus a constant. It is especially useful for solving equations that cannot be factored easily.

配方法是一种强大的技巧，它将二次方程转化为一个完全平方三项式加上一个常数。这对于解那些不易因式分解的方程特别有用。

The general procedure for solving x² + bx + c = 0:

解 x² + bx + c = 0 的一般步骤：

Move the constant term to the right side: x² + bx = -c.
将常数项移到右边：x² + bx = -c。
Add (b/2)² to both sides: x² + bx + (b/2)² = (b/2)² – c.
两边加上 (b/2)²：x² + bx + (b/2)² = (b/2)² – c。
Write the left side as a perfect square: (x + b/2)² = (b/2)² – c.
将左边写成完全平方：(x + b/2)² = (b/2)² – c。
Take the square root of both sides and solve for x.
两边开平方根，解出 x。

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

示例：用配方法解 x² + 6x – 7 = 0。

Here b = 6, so (b/2)² = 9. Add 9 to both sides: x² + 6x + 9 = 7 + 9, hence (x + 3)² = 16. Taking square roots gives x + 3 = ±4, so x = 1 or x = -7.

这里 b = 6，所以 (b/2)² = 9。两边加9：x² + 6x + 9 = 7 + 9，即 (x + 3)² = 16。开平方根得到 x + 3 = ±4，所以 x = 1 或 x = -7。

This method also helps in deriving the quadratic formula and in finding the vertex of a parabola.

这种方法也有助于推导二次公式以及找到抛物线的顶点。

4. Quadratic Formula | 二次公式

The quadratic formula works for any quadratic equation, regardless of whether it factors neatly. It is derived from completing the square on the general equation ax² + bx + c = 0.

二次公式适用于任何二次方程，无论它能否整齐地因式分解。它是通过对一般方程 ax² + bx + c = 0 配方推导出来的。

x = (-b ± √(b² – 4ac)) / (2a)

To use the formula, substitute the coefficients a, b, and c directly. Remember that the symbol ± means there are two solutions: one with a plus sign and one with a minus sign.

使用公式时，直接代入系数 a、b 和 c。记住符号 ± 表示有两个解：一个用加号，一个用减号。

Example: Solve 3x² – 5x – 2 = 0 using the quadratic formula.

示例：使用二次公式解 3x² – 5x – 2 = 0。

Here a = 3, b = -5, c = -2. Substitute into the formula:

这里 a = 3，b = -5，c = -2。代入公式：

x = (5 ± √(25 – 4×3×(-2))) / (6) = (5 ± √(25+24)) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6

So x = (5+7)/6 = 2 or x = (5-7)/6 = -1/3.

所以 x = (5+7)/6 = 2 或 x = (5-7)/6 = -1/3。

Always check the discriminant b² – 4ac first to know if real solutions exist.

在使用公式前，总是先检查判别式 b² – 4ac 以确定是否存在实数解。

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 fully.

判别式是二次公式中根号下的部分：Δ = b² – 4ac。它告诉我们根的性质，而无需完整解方程。

For a quadratic equation ax² + bx + c = 0:

对于二次方程 ax² + bx + c = 0：

If Δ > 0, there are two distinct real roots.
如果 Δ > 0，有两个不同的实数根。
If Δ = 0, there is exactly one real root (a repeated root).
如果 Δ = 0，恰好有一个实数根（重根）。
If Δ < 0, there are no real roots; the solutions are complex numbers.
如果 Δ < 0，没有实数根；解是复数。

Example: For 2x² – 4x + 2 = 0, Δ = 16 – 16 = 0, so the equation has one repeated root (x = 1).

示例：对于 2x² – 4x + 2 = 0，Δ = 16 – 16 = 0，因此方程有一个重根（x = 1）。

The discriminant also helps in sketching graphs and solving inequalities involving quadratics.

判别式还有助于绘制图形以及求解涉及二次函数的不等式。

6. Graphical Interpretation | 图形解释

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

二次函数 y = ax² + bx + c 的图形是一条抛物线。方程 ax² + bx + c = 0 的根是抛物线与 x 轴交点的横坐标。

If the parabola crosses the x-axis at two points, the equation has two distinct real roots.
如果抛物线在两个点穿过 x 轴，则方程有两个不同的实数根。
If the vertex touches the x-axis, there is one repeated root.
如果顶点与 x 轴相切，则有一个重根。
If the parabola lies entirely above or below the x-axis, there are no real roots.
如果抛物线完全位于 x 轴上方或下方，则没有实数根。

The vertex of the parabola is located at x = -b/(2a). Substituting this x-value gives the minimum or maximum value of the function, depending on the sign of a.

抛物线的顶点位于 x = -b/(2a)。将这个 x 值代入得到函数的最小值或最大值，具体取决于 a 的符号。

Example: y = x² – 2x – 3 has roots at x = -1 and x = 3, and its vertex is at x = 1, giving minimum y = -4.

示例：y = x² – 2x – 3 的根为 x = -1 和 x = 3，顶点在 x = 1 处，最小值为 y = -4。

7. Solving Word Problems | 应用题求解

Quadratic equations frequently appear in real-world contexts such as geometry, motion, and business. The key is to translate the problem into an algebraic equation and then solve it using an appropriate method.

二次方程经常出现在几何、运动、商业等现实情境中。关键是将其转化为代数方程，然后使用合适的方法求解。

Steps to solve a word problem:

解应用题的步骤：

Read the problem carefully and identify the unknown quantity.
仔细阅读问题，识别未知量。
Form a quadratic equation using the given information.
利用给定信息建立二次方程。
Solve the equation (factorize, complete the square, or use the formula).
解方程（因式分解、配方或使用公式）。
Check whether the answers make sense in the context of the problem (e.g., length cannot be negative).
检查答案在问题情境中是否合理（例如，长度不能为负数）。

Example: The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Find the dimensions.

示例：一个矩形的长比宽多 3 厘米，面积为 40 平方厘米。求其尺寸。

Let width = w, then length = w + 3. The area equation is w(w + 3) = 40, so w² + 3w – 40 = 0. Factoring gives (w + 8)(w – 5) = 0, so w = -8 (reject) or w = 5. Therefore width = 5 cm, length = 8 cm.

设宽 = w，则长 = w + 3。面积方程为 w(w + 3) = 40，即 w² + 3w – 40 = 0。因式分解得 (w + 8)(w – 5) = 0，所以 w = -8（舍去）或 w = 5。因此宽 = 5 厘米，长 = 8 厘米。

8. Common Mistakes and Tips | 常见错误与技巧

Students often lose marks on quadratic equations due to avoidable errors. Here are some common ones and how to avoid them:

学生常常因为可避免的错误而在二次方程上失分。以下是一些常见错误及如何避免它们：

Forgetting to write the equation in the form ax² + bx + c = 0 before factorizing. Always rearrange first.
忘记在因式分解前将方程写成 ax² + bx + c = 0 的形式。 始终先整理方程。
Misusing the quadratic formula signs. Carefully substitute negative values of b and c.
错误使用二次公式的符号。 小心代入 b 和 c 的负值。
Assuming there is always a real solution. Check the discriminant first.
假设总是有实数解。 先检查判别式。
Forgetting to reject invalid solutions in word problems. For example, negative lengths or times.
在应用题中忘记舍去无效解。 例如，负的长度或时间。

Quick tips: When factorizing, always expand your factors to check; when using the formula, write down every substitution step; when completing the square, remember to add the same value to both sides.

快速技巧：因式分解时，总是展开你的因子进行检查；使用公式时，写下每一个代入步骤；配方时，记住两边加同一个值。

9. Practice Questions | 练习问题

Try these questions to test your understanding. Solutions are provided in abbreviated form.

尝试以下问题来测试你的理解。附有简要答案。

Solve by factorizing: x² – 9x + 20 = 0. Answer: x = 4 or x = 5
用因式分解法求解：x² – 9x + 20 = 0。 答案：x = 4 或 x = 5
Solve by completing the square: x² + 4x – 12 = 0. Answer: x = 2 or x = -6
用配方法求解：x² + 4x – 12 = 0。 答案：x = 2 或 x = -6
Use the quadratic formula to solve 2x² – 3x – 5 = 0. Answer: x = 2.5 or x = -1
使用二次公式求解 2x² – 3x – 5 = 0。 答案：x = 2.5 或 x = -1
Determine the number of real roots of 4x² – 4x + 1 = 0. Answer: Δ = 0, one repeated root.
确定 4x² – 4x + 1 = 0 的实数根个数。 答案：Δ = 0，一个重根。

Practising more problems will help you recognise patterns and choose the best solving strategy quickly.

多练习将帮助你识别模式，并快速选择最佳求解策略。

10. Summary | 总结

Quadratic equations are a key part of IGCSE Mathematics. You should be comfortable with the standard form, the methods of solving (factorization, completing the square, and the quadratic formula), and the interpretation of the discriminant and graphs.

二次方程是IGCSE数学的关键部分。你应该熟悉标准形式、求解方法（因式分解、配方和二次公式），以及判别式和图形的解释。

Remember: always check the discriminant first, choose the most efficient method, and verify your solutions in the original equation, especially in word problems.

记住：总是先检查判别式，选择最有效的方法，并在原方程中验证你的解，特别是在应用题中。

Mastering quadratics will also prepare you for higher-level topics such as functions, inequalities, and calculus.

掌握二次方程也将为更高层次的主题做好准备，如函数、不等式和微积分。

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Solving Quadratic Equations | 解二次方程