Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in the IGCSE Mathematics syllabus. It appears in both Foundation and Higher Tier papers for Edexcel, and a strong understanding of its methods is essential for success. In this revision guide, we will explore all the standard techniques for solving quadratic equations, along with worked examples, common pitfalls, and practice strategies.

二次方程是 IGCSE 数学大纲中最重要的内容之一。在 Edexcel 的 Foundation 和 Higher Tier 试卷中都会出现，深入理解其解法是取得高分的关键。在本复习指南中，我们将探讨解二次方程的所有标准方法，包括例题、常见错误以及练习策略。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The highest power of the variable x is 2. Examples include x² + 3x – 4 = 0, 2x² – 5x + 1 = 0, and 9x² – 16 = 0.

二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。变量 x 的最高次数为 2。例如 x² + 3x – 4 = 0、2x² – 5x + 1 = 0、9x² – 16 = 0 都是二次方程。

If the equation is not given in this standard form, you must first rearrange it so that all terms are on one side and the right-hand side is 0. This is a crucial first step before applying any solving method.

如果方程未以标准形式给出，你必须先重新整理，使所有项在等号一侧，右侧为 0。这是应用任何解法前至关重要的一步。

2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic expression can be factorised into two linear factors. The principle is that if the product of two expressions is zero, then at least one of them must be zero.

当二次表达式可以分解为两个一次因式时，因式分解法是最快捷的方法。其原理是：如果两个表达式的乘积为零，那么至少其中一个必须为零。

For example, solve x² – 5x + 6 = 0.

例如，解方程 x² – 5x + 6 = 0。

Step 1: Factorise the left-hand side. We look for two numbers that multiply to 6 and add to -5. These are -2 and -3.

步骤 1： 将左侧因式分解。寻找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3。

x² – 5x + 6 = (x – 2)(x – 3) = 0

Step 2: Set each factor equal to zero and solve:

步骤 2： 令每个因式等于零并求解：

x – 2 = 0 or x – 3 = 0

x = 2 or x = 3

So the solutions are x = 2 and x = 3.

因此解为 x = 2 和 x = 3。

When a ≠ 1, such as 2x² + 5x + 2 = 0, factorisation requires a different pattern. Multiply the coefficient of x² (2) by the constant term (2) to get 4. Find two numbers whose product is 4 and sum is 5: they are 1 and 4. Then rewrite the middle term and factorise by grouping.

当 a ≠ 1 时，例如 2x² + 5x + 2 = 0，因式分解需要另一种模式。将 x² 的系数（2）与常数项（2）相乘得到 4。寻找乘积为 4 且和为 5 的两个数：它们是 1 和 4。然后重新改写中间项，并分组因式分解。

2x² + 5x + 2 = 2x² + x + 4x + 2 = x(2x + 1) + 2(2x + 1) = (2x + 1)(x + 2)

Thus (2x + 1)(x + 2) = 0, giving x = -½ or x = -2.

因此 (2x + 1)(x + 2) = 0，得 x = -½ 或 x = -2。

3. Completing the Square | 配方法

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

配方法将二次式改写为 a(x + p)² + q 的形式。当因式分解困难或无法进行时，这种方法特别适用于求抛物线的顶点以及解方程。

For example, solve x² + 6x – 7 = 0 by completing the square.

例如，用配方法解 x² + 6x – 7 = 0。

Step 1: Move the constant term to the other side.

步骤 1： 将常数项移到等号另一侧。

x² + 6x = 7

Step 2: Take half of the coefficient of x (which is 6), square it, and add to both sides.

步骤 2： 取 x 的系数（6）的一半，平方，然后加到等式两侧。

x² + 6x + (3)² = 7 + 9

(x + 3)² = 16

Step 3: Take the square root of both sides, remembering the ± sign.

步骤 3： 对两侧取平方根，记得加上 ± 号。

x + 3 = ±4

x = -3 + 4 or x = -3 – 4

x = 1 or x = -7

When the coefficient of x² is not 1, factor it out first. For example, 2x² – 8x + 1 = 0 becomes 2(x² – 4x) + 1 = 0. Complete the square inside the bracket: (x – 2)² – 4. Thus the expression becomes 2[(x – 2)² – 4] + 1 = 2(x – 2)² – 8 + 1 = 2(x – 2)² – 7.

当 x² 的系数不为 1 时，先将其提取出来。例如，2x² – 8x + 1 = 0 变为 2(x² – 4x) + 1 = 0。在括号内配方：(x – 2)² – 4。因此原式变为 2[(x – 2)² – 4] + 1 = 2(x – 2)² – 8 + 1 = 2(x – 2)² – 7。

4. The Quadratic Formula | 二次公式

The quadratic formula solves any quadratic equation, provided that you substitute the values of a, b and c correctly. This formula is given on the Edexcel IGCSE formula sheet, but you must know how to use it.

二次公式可以解任何二次方程，前提是正确代入 a、b、c 的值。该公式在 Edexcel IGCSE 公式表中给出，但你必须知道如何使用它。

x = (-b ± √(b² – 4ac)) / (2a)

For example, solve 2x² – 3x – 2 = 0 using the formula.

例如，用公式解 2x² – 3x – 2 = 0。

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

这里 a = 2, b = -3, c = -2。代入公式：

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

x = (3 ± √(9 + 16)) / 4

x = (3 ± √25) / 4 = (3 ± 5) / 4

x = 2 or x = -½

Always write down the values of a, b and c before substituting. This reduces errors, especially when b is negative.

代入前总是先写下 a、b、c 的值。这能减少错误，尤其是当 b 为负数时。

5. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: b² – 4ac. It tells you the number and type of solutions without fully solving the equation.

判别式是二次公式中根号内的部分：b² – 4ac。它不需要完全解方程就能告诉我们解的数量和类型。

If b² – 4ac > 0, there are two distinct real roots.
如果 b² – 4ac > 0，则有两个不相等的实数根。
If b² – 4ac = 0, there is one repeated real root.
如果 b² – 4ac = 0，则有一个相等的实数根（重根）。
If b² – 4ac < 0, there are no real roots (only complex roots).
如果 b² – 4ac < 0，则没有实数根（只有复数根）。

For example, determine the nature of the roots of x² – 4x + 4 = 0.

例如，判断 x² – 4x + 4 = 0 的根的性质。

Δ = (-4)² – 4 × 1 × 4 = 16 – 16 = 0

Since Δ = 0, the equation has one repeated real root: x = 2 (double root).

由于 Δ = 0，方程有一个相等的实数根：x = 2（二重根）。

6. Solving Quadratic Equations by Factorising a Difference of Two Squares | 用平方差公式因式分解

Some quadratics have only two terms, such as x² – 9 = 0. This is a difference of two squares and can be factorised as (x – 3)(x + 3) = 0, giving x = 3 or x = -3. The general form is a² – b² = (a – b)(a + b).

有些二次方程只有两项，例如 x² – 9 = 0。这是平方差形式，可以分解为 (x – 3)(x + 3) = 0，得到 x = 3 或 x = -3。一般形式为 a² – b² = (a – b)(a + b)。

Another example: 4x² – 25 = 0. Here (2x)² – 5² = 0, so (2x – 5)(2x + 5) = 0. Therefore x = 5/2 or x = -5/2.

另一个例子：4x² – 25 = 0。这里 (2x)² – 5² = 0，所以 (2x – 5)(2x + 5) = 0。因此 x = 5/2 或 x = -5/2。

Remember to solve for x even if the equation looks simple. Some students forget to take the square root of both sides when using this method.

请记住，即使方程看起来简单，也要解出 x。有些学生在使用这种方法时忘记对两侧开平方根。

7. Solving Quadratic Equations by Using the Graph | 图像法解二次方程

You can also solve a quadratic equation by drawing its graph. The solutions are the x-coordinates where the curve y = ax² + bx + c crosses the x-axis (i.e., where y = 0).

你也可以通过绘制二次函数的图像来解二次方程。解是曲线 y = ax² + bx + c 与 x 轴交点的 x 坐标（即 y = 0 处）。

For example, if the graph of y = x² – 2x – 3 crosses the x-axis at x = -1 and x = 3, then the solutions to x² – 2x – 3 = 0 are x = -1 and x = 3.

例如，如果 y = x² – 2x – 3 的图像与 x 轴交于 x = -1 和 x = 3，那么方程 x² – 2x – 3 = 0 的解是 x = -1 和 x = 3。

If the graph does not touch the x-axis, the equation has no real roots. If it touches at just one point, there is a repeated root. This method is less accurate than algebraic methods unless the roots are integer values, but it is often used to estimate solutions.

如果图像不与 x 轴相交，则方程没有实数根。如果只接触于一点，则有一个重根。除非根为整数，否则图像法不如代数方法精确，但常用于估算解。

8. Word Problems Involving Quadratic Equations | 二次方程应用题

Quadratic equations often arise in geometry, number problems and projectile motion. The key is to translate the problem into an equation, solve it, and then check that the solutions make sense in the context.

二次方程通常出现在几何、数问题和抛体运动中。关键是将其转化为方程，求解，然后检查解在上下文中是否有意义。

Example: The area of a rectangle is 24 cm². Its length is 5 cm more than its width. Find the width.

例：一个矩形的面积为 24 cm²。它的长比宽多 5 cm。求宽。

Let the width be w. Then the length is w + 5. The area is w(w + 5) = 24.

设宽为 w，则长为 w + 5。面积是 w(w + 5) = 24。

w² + 5w – 24 = 0

Factorise: (w + 8)(w – 3) = 0, so w = -8 or w = 3. Since a width cannot be negative, w = 3 cm.

分解：(w + 8)(w – 3) = 0，所以 w = -8 或 w = 3。由于宽度不能为负，故 w = 3 cm。

Always state the units in your final answer and reject any solution that is not physically possible.

在最终答案中始终写出单位，并舍去任何不符合实际的解。

9. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Below are some frequent errors made by students, along with tips to avoid them.

以下是学生常犯的一些错误，以及避免这些错误的小贴士。

Not rearranging the equation to zero: Always write the equation in the form ax² + bx + c = 0 before factorising or using the formula.
未将方程整理为零： 在因式分解或使用公式前，始终写成 ax² + bx + c = 0 的形式。
Forgetting the ± sign: When taking the square root of both sides, always remember that there are two possible values (unless the discriminant is zero).
忘记 ± 号： 对两侧取平方根时，始终记住有两个可能的值（除非判别式为零）。
Sign errors in substitution: In the quadratic formula, replace b by its exact value including the negative sign. Use brackets when substituting.
代入时符号错误： 在二次公式中，将 b 替换为其确切值，包括负号。代入时使用括号。
Dividing both sides by x: Never divide by x when x = 0 might be a solution. This can lose a root. Instead, factorise the expression.
两边同时除以 x： 当 x = 0 可能是解时，绝不要两边除以 x。这可能会丢失一个根。而应因式分解表达式。

10. Practice Questions | 练习题

Try these questions on your own, then check the answers below.

尝试独立完成以下问题，然后对照下方答案检查。

Question 1: Solve x² – 7x + 12 = 0 by factorisation.

问题 1： 用因式分解法解 x² – 7x + 12 = 0。

Question 2: Solve 3x² + 2x – 1 = 0 using the quadratic formula.

问题 2： 用二次公式解 3x² + 2x – 1 = 0。

Question 3: Find the value of k such that the equation x² + kx + 9 = 0 has a repeated root.

问题 3： 求 k 的值，使得方程 x² + kx + 9 = 0 具有重根。

Question 4: A number increased by its reciprocal is equal to 2. Find the number.

问题 4： 一个数与其倒数之和等于 2。求这个数。

Answers:

答案：

1: x = 3 or x = 4.
1 答案： x = 3 或 x = 4。
2: x = 1/3 or x = -1.
2 答案： x = 1/3 或 x = -1。
3: For a repeated root, the discriminant must be 0, so k² – 36 = 0, giving k = 6 or k = -6.
3 答案： 对于重根，判别式必须为 0，所以 k² – 36 = 0，得 k = 6 或 k = -6。
4: Let the number be n. Then n + 1/n = 2. Multiply by n: n² + 1 = 2n, so n² – 2n + 1 = 0, i.e., (n – 1)² = 0, giving n = 1.
4 答案： 设数为 n，则 n + 1/n = 2。两边乘以 n：n² + 1 = 2n，所以 n² – 2n + 1 = 0，即 (n – 1)² = 0，得 n = 1。

11. Summary | 总结

To solve a quadratic equation, follow these steps:

解二次方程，请遵循以下步骤：

Rearrange the equation into the standard form ax² + bx + c = 0.
将方程整理为标准形式 ax² + bx + c = 0。
Try factorisation first. If the expression can be factorised, it is usually the quickest method.
首先尝试因式分解。如果表达式可以分解，通常是最快的方法。
If factorisation is not possible, use the quadratic formula or completing the square.
如果因式分解不行，使用二次公式或配方法。
Use the discriminant to check the number of real roots before solving.
在求解前使用判别式检查实数根的个数。
In word problems, interpret your solutions and discard any that do not fit the context.
在应用题中，解释你的解并舍弃不符合上下文的解。

With consistent practice, solving quadratic equations will become second nature. Good luck with your revision!

通过持续练习，解二次方程将变得得心应手。祝复习顺利！

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Solving Quadratic Equations | 解二次方程