Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in the Edexcel IGCSE Mathematics syllabus. Mastering them is essential for success in both foundation and higher tier exams. This article covers all main methods of solving quadratics, important algebraic techniques, graph sketching, and common pitfalls.

二次方程在 Edexcel IGCSE 数学考纲中频繁出现。掌握二次方程对基础级和高级别考试都至关重要。本文将全面讲解解二次方程的主要方法、关键代数技巧、图像绘制以及常见易错点。

1. Expanding and Factorising Quadratics | 展开与因式分解二次式

A quadratic expression is usually written in the form ax² + bx + c, where a, b, c are constants and a ≠ 0. Expanding means removing brackets, while factorising is the reverse process of writing the expression as a product of two linear factors.

二次表达式通常写成 ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。展开是指去掉括号，而因式分解是逆过程，即将表达式写成两个一次因式的乘积。

For example, expand (x + 3)(x + 5) = x² + 8x + 15. To factorise x² + 8x + 15, we look for two numbers that multiply to 15 and add to 8. Those numbers are 3 and 5, so x² + 8x + 15 = (x + 3)(x + 5).

例如，展开 (x + 3)(x + 5) = x² + 8x + 15。要将 x² + 8x + 15 因式分解，我们寻找两个数相乘得 15 且相加得 8。这两个数是 3 和 5，所以 x² + 8x + 15 = (x + 3)(x + 5)。

When a ≠ 1, factorising becomes more involved. For instance, 2x² + 7x + 3 = (2x + 1)(x + 3). Always check your answer by expanding.

当 a ≠ 1 时，因式分解更复杂。例如，2x² + 7x + 3 = (2x + 1)(x + 3)。一定要通过展开来检查答案。

2. Solving by Factorising | 因式分解法解二次方程

The key principle is: if a product of two factors is zero, then at least one of the factors must be zero. So if (x – 2)(x + 5) = 0, then x – 2 = 0 or x + 5 = 0, giving x = 2 or x = -5.

关键原理是：如果两个因式的乘积为零，则至少有一个因式为零。所以如果 (x – 2)(x + 5) = 0，那么 x – 2 = 0 或 x + 5 = 0，解得 x = 2 或 x = -5。

Steps for solving a quadratic equation by factorising:

用因式分解法解二次方程的步骤：

Rearrange the equation so that one side is zero. | 重新整理方程，使一边为零。
Factorise the quadratic expression completely. | 将二次表达式完全因式分解。
Set each factor equal to zero and solve. | 令每个因式等于零并求解。

Example: Solve x² – 3x – 10 = 0. Factoring gives (x – 5)(x + 2) = 0. Hence x = 5 or x = -2.

示例：解 x² – 3x – 10 = 0。因式分解得 (x – 5)(x + 2) = 0。因此 x = 5 或 x = -2。

Be careful when the coefficient of x² is not 1. For example, 6x² + 5x – 6 = 0 can be factorised as (3x – 2)(2x + 3) = 0, giving x = 2/3 or x = -3/2.

当 x² 系数不为 1 时要小心。例如，6x² + 5x – 6 = 0 可因式分解为 (3x – 2)(2x + 3) = 0，得到 x = 2/3 或 x = -3/2。

3. Solving by Completing the Square | 配方法解二次方程

Completing the square rewrites a quadratic expression in the form p(x + q)² + r. This method works for any quadratic equation, even when factorisation is not possible.

配方法将二次表达式改写为 p(x + q)² + r 的形式。这种方法适用于任何二次方程，即使无法因式分解也一样可行。

For a quadratic x² + bx + c, we use the identity:

对于二次式 x² + bx + c，我们使用恒等式：

x² + bx + c = (x + b/2)² – (b/2)² + c

Example: Solve x² + 6x + 2 = 0 by completing the square.

示例：用配方法解 x² + 6x + 2 = 0。

x² + 6x = -2

Add 9 to both sides because (6/2)² = 9:

两边加上 9，因为 (6/2)² = 9：

x² + 6x + 9 = 7

(x + 3)² = 7

Take square roots: x + 3 = ±√7, so x = -3 ± √7.

开平方：x + 3 = ±√7，所以 x = -3 ± √7。

When the leading coefficient is not 1, factor it out first before completing the square. For example, 2x² + 8x + 5 = 2(x² + 4x) + 5, then complete the square inside the bracket.

当最高次项系数不为 1 时，先将其提出，再进行配方。例如，2x² + 8x + 5 = 2(x² + 4x) + 5，然后在括号内配方。

4. The Quadratic Formula | 二次求根公式

The quadratic formula is a universal method for solving any quadratic equation ax² + bx + c = 0. The formula is derived from completing the square and is given by:

二次求根公式是解任意二次方程 ax² + bx + c = 0 的通用方法。该公式由配方法推导而来，表达式为：

x = (-b ± √(b² – 4ac)) / (2a)

To use the formula, identify a, b, and c from the equation, substitute them carefully, and simplify. Remember to use the ± symbol to obtain two solutions.

使用公式时，从方程中识别 a、b、c，仔细代入并化简。记得用 ± 符号得到两个解。

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

示例：用二次公式解 2x² – 4x – 3 = 0。

x = (4 ± √((-4)² – 4 × 2 × (-3))) / (2 × 2)

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

This simplifies to x = 1 ± (√10)/2. Note that √40 = 2√10, so (4 ± 2√10)/4 = 1 ± √10/2.

这化简为 x = 1 ± (√10)/2。注意 √40 = 2√10，所以 (4 ± 2√10)/4 = 1 ± √10/2。

It is important to simplify surds when possible, as IGCSE examiners expect fully simplified answers.

尽可能化简根式很重要，因为 IGCSE 考官期待完全化简的答案。

5. The Discriminant | 判别式

The expression b² – 4ac inside the quadratic formula is called the discriminant. It tells us how many real roots a quadratic equation has, without solving it fully.

二次公式中的表达式 b² – 4ac 称为判别式。它告诉我们二次方程有多少个实数根，而无需完全求解。

Discriminant value	Number of real roots	Graph interpretation
b² – 4ac > 0	Two distinct real roots	The parabola crosses the x-axis twice.
b² – 4ac = 0	One repeated real root	The parabola touches the x-axis at one point.
b² – 4ac < 0	No real roots	The parabola does not intersect the x-axis.

For example, for the equation x² – 4x + 4 = 0, the discriminant is (-4)² – 4 × 1 × 4 = 0, so there is exactly one real root, x = 2.

例如，对于方程 x² – 4x + 4 = 0，判别式为 (-4)² – 4 × 1 × 4 = 0，所以正好有一个实数根，x = 2。

For the equation x² + 2x + 5 = 0, the discriminant is 4 – 20 = -16 < 0, so there are no real roots.

对于方程 x² + 2x + 5 = 0，判别式为 4 – 20 = -16 < 0，所以没有实数根。

6. Sketching Quadratic Graphs | 绘制二次函数图像

A quadratic function y = ax² + bx + c always produces a parabola. The sign of a determines whether the parabola opens upward (a > 0) or downward (a < 0).

二次函数 y = ax² + bx + c 的图像总是抛物线。a 的符号决定抛物线开口向上（a > 0）还是向下（a < 0）。

To sketch a quadratic graph, you need three key features:

绘制二次函数草图需要三个关键特征：

Roots: The x-values where y = 0. These are the solutions of ax² + bx + c = 0. | 根：使 y = 0 的 x 值，即方程 ax² + bx + c = 0 的解。
y-intercept: The point (0, c) where the graph crosses the y-axis. | y 轴截距：图像与 y 轴交于点 (0, c)。
Vertex (turning point): The maximum or minimum point, located at x = -b/(2a). Substitute this x-value to find the y-coordinate. | 顶点（转向点）：最高点或最低点，位于 x = -b/(2a)。将 x 值代入求得 y 坐标。

Example: Sketch y = x² – 2x – 3. | 示例：绘制 y = x² – 2x – 3 的草图。

The roots are found by solving x² – 2x – 3 = 0, which factors as (x – 3)(x + 1) = 0, so x = 3 and x = -1. The y-intercept is (0, -3). The vertex occurs at x = -b/(2a) = 2/2 = 1, and y = 1 – 2 – 3 = -4. So the vertex is (1, -4).

通过解 x² – 2x – 3 = 0 求根，因式分解为 (x – 3)(x + 1) = 0，所以 x = 3 和 x = -1。y 轴截距为 (0, -3)。顶点在 x = -b/(2a) = 2/2 = 1，y = 1 – 2 – 3 = -4。所以顶点为 (1, -4)。

7. Applications and Word Problems | 应用与文字题

Quadratic equations are used to model real-life situations such as projectile motion, area problems, and profit calculations. In IGCSE examinations, you must translate a word problem into a quadratic equation and then solve it.

二次方程常用于建模现实生活情境，例如抛体运动、面积问题以及利润计算。在 IGCSE 考试中，你需要将文字题转化为二次方程再求解。

Typical example: A rectangle has length (x + 3) cm and width (x – 2) cm. Its area is 24 cm². Find the value of x.

典型示例：一个矩形的长为 (x + 3) cm，宽为 (x – 2) cm。面积为 24 cm²。求 x 的值。

Set up the equation: (x + 3)(x – 2) = 24. Expanding gives x² + x – 6 = 24, so x² + x – 30 = 0. Factorising: (x + 6)(x – 5) = 0, so x = -6 or x = 5. Since length and width must be positive, reject x = -6. Hence x = 5.

建立方程：(x + 3)(x – 2) = 24。展开得 x² + x – 6 = 24，所以 x² + x – 30 = 0。因式分解：(x + 6)(x – 5) = 0，所以 x = -6 或 x = 5。由于长和宽必须为正，舍去 x = -6。因此 x = 5。

When setting up equations, always define the variable clearly and check whether each solution makes sense in the context of the problem.

建立方程时，务必清晰定义变量，并检查每个解在问题情境中是否合理。

8. Common Mistakes and Tips | 常见错误与技巧

Many students lose marks on quadratic equations due to avoidable errors. Here are the most common ones to watch out for.

许多学生因可避免的错误在二次方程题目中失分。以下是最需要注意的常见错误。

Forgetting to set the equation to zero before factorising. | 因式分解前忘记将方程化为零。
Incorrect signs when extracting roots from both sides. Remember x² = a gives x = ±√a, not just +√a. | 两边开平方时符号错误。记住 x² = a 给出 x = ±√a，而不仅仅是 +√a。
Misidentifying coefficients a, b, c in the quadratic formula, especially when c is negative. | 在二次公式中错误识别系数 a、b、c，尤其是当 c 为负数时。
Not simplifying surds completely. | 未完全化简根式。
Dropping a solution when solving by factorisation. There are always two solutions (or one repeated solution). | 因式分解法求解时遗漏一个解。总是有两个解（或一个重根）。

Always check your solutions by substituting them back into the original equation. This only takes a few seconds but can prevent careless mistakes.

始终将解代回原方程验证。这只需几秒钟，但能避免粗心错误。

9. Practice Questions | 练习题

Try these problems to consolidate your understanding. Answers are provided after each question.

尝试以下问题来巩固理解。每个问题后附有答案。

1. Solve 3x² – 11x – 4 = 0 by factorising. | 1. 用因式分解法解 3x² – 11x – 4 = 0。

Answer: (3x + 1)(x – 4) = 0, so x = -1/3 or x = 4.

答案：(3x + 1)(x – 4) = 0，所以 x = -1/3 或 x = 4。

2. Complete the square for x² – 8x + 5 and solve x² – 8x + 5 = 0.

2. 将 x² – 8x + 5 配方，并解 x² – 8x + 5 = 0。

Answer: (x – 4)² – 11 = 0, so x = 4 ± √11.

答案：(x – 4)² – 11 = 0，所以 x = 4 ± √11。

3. Use the quadratic formula to solve 5x² + 2x – 1 = 0, giving your answers correct to 2 decimal places.

3. 用二次公式解 5x² + 2x – 1 = 0，答案精确到两位小数。

Answer: x = (-2 ± √(4 + 20)) / 10 = (-2 ± √24) / 10 ≈ 0.29 or -0.69.

答案：x = (-2 ± √(4 + 20)) / 10 = (-2 ± √24) / 10 ≈ 0.29 或 -0.69。

4. Find the discriminant of x² + 3x + 7 = 0 and state the number of real roots.

4. 求 x² + 3x + 7 = 0 的判别式并说明实数根的个数。

Answer: Discriminant = 9 – 28 = -19 < 0, so no real roots.

答案：判别式 = 9 – 28 = -19 < 0，所以没有实数根。

5. The height h (in metres) of a ball thrown upward is given by h = 2 + 9t – 5t², where t is time in seconds. When does the ball hit the ground?

5. 向上抛出的球的高度 h（米）由 h = 2 + 9t – 5t² 给出，其中 t 为时间（秒）。球何时落地？

Answer: Set h = 0, so 5t² – 9t – 2 = 0. Using the quadratic formula, t = (9 ± √(81 + 40)) / 10 = (9 ± √121) / 10 = (9 ± 11) / 10, giving t = 2 or t = -0.2. Time cannot be negative, so t = 2 seconds.

答案：令 h = 0，得 5t² – 9t – 2 = 0。用二次公式，t = (9 ± √(81 + 40)) / 10 = (9 ± √121) / 10 = (9 ± 11) / 10，得 t = 2 或 t = -0.2。时间不能为负，所以 t = 2 秒。

10. Summary | 总结

Quadratic equations can be solved by factorisation, completing the square, or using the quadratic formula. The discriminant helps determine the nature of the roots. Being able to sketch graphs and apply quadratics to real-world problems is essential for the IGCSE examination.

二次方程可以通过因式分解法、配方法或二次求根公式求解。判别式有助于判断根的性质。能够绘制图像并将二次方程应用于实际问题，对于 IGCSE 考试至关重要。

Regular practice and careful checking of your working will build confidence. Remember that every quadratic equation has at most two real roots, and the method you choose should be the one you are most comfortable with, while also being efficient for the given equation.

经常练习并仔细检查过程可增强自信心。记住每个二次方程最多有两个实数根，选择的方法应当是你最熟练且对给定方程最有效率的一种。

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Solving Quadratic Equations | 解二次方程