Solving Quadratic Equations | 二次方程求解

📚 Solving Quadratic Equations | 二次方程求解

Quadratic equations are one of the most important topics in the IGCSE Mathematics syllabus. In this comprehensive revision guide, we will explore the standard form of quadratic equations, multiple solving methods, the discriminant, and real-world applications. Whether you are preparing for Paper 2 or Paper 4, mastering this topic is essential.

二次方程是 IGCSE 数学课程中最重要的主题之一。在本综合复习指南中，我们将探讨二次方程的标准形式、多种求解方法、判别式以及实际应用。无论你是在准备 Paper 2 还是 Paper 4，掌握这一主题都至关重要。

1. What Are Quadratic Equations? | 什么是二次方程

A quadratic equation is a polynomial equation of degree 2. The general form of a quadratic equation in one variable \(x\) is written as ax² + bx + c = 0, where a, b, and c are constants, a ≠ 0, and x is the variable. The coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term.

二次方程是次数为 2 的多项式方程。含一个变量 x 的二次方程的一般形式写作 ax² + bx + c = 0，其中 a、b、c 是常数，a ≠ 0，x 是变量。系数 a 称为首项系数，b 称为一次项系数，c 称为常数项。

The highest power of the variable is 2, which gives the equation its name “quadratic” (from the Latin word “quadratus”, meaning square). The solutions of a quadratic equation are also called roots or zeros of the equation.

变量的最高次数是 2，这赋予了方程”二次”的名称（源自拉丁语”quadratus”，意为平方）。二次方程的解也称为方程的根或零点。

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

A quadratic equation must be rearranged into standard form ax² + bx + c = 0 before solving. This is because all standard solving methods — factorisation, completing the square, and the quadratic formula — require the equation to equal zero on one side.

在求解之前，必须将二次方程化为标准形式 ax² + bx + c = 0。这是因为所有标准解法——因式分解法、配方法和二次公式法——都要求方程的一边等于零。

For example, the equation 2x² + 5x = 3 is not in standard form. We rearrange it by subtracting 3 from both sides to obtain 2x² + 5x − 3 = 0. Here, a = 2, b = 5, and c = −3.

例如，方程 2x² + 5x = 3 不处于标准形式。我们通过两边同时减去 3 来整理它，得到 2x² + 5x − 3 = 0。此时，a = 2，b = 5，c = −3。

The roots of a quadratic equation represent the x-values where the corresponding quadratic function crosses the x-axis. If the equation has no real roots, the graph does not intersect the x-axis.

二次方程的根表示对应二次函数与 x 轴相交处的 x 值。如果方程没有实数根，则其图像不与 x 轴相交。

3. Solving by Factorisation | 因式分解法求解

Factorisation is the first method taught for solving quadratic equations, and it is often the fastest. The principle is simple: if the product of two expressions equals zero, then at least one of the expressions must equal zero. This is known as the zero product property.

因式分解法是学习求解二次方程时最先教授的方法，也往往是最快捷的方法。其原理很简单：如果两个表达式的乘积等于零，那么至少其中一个表达式必须等于零。这称为零乘积性质。

To solve a quadratic equation of the form x² + bx + c = 0 by factorisation, we look for two numbers whose product is c and whose sum is b. For example, to solve x² + 7x + 12 = 0, we find two numbers whose product is 12 and sum is 7. These numbers are 3 and 4.

要通过因式分解法求解形如 x² + bx + c = 0 的二次方程，我们寻找两个数，其乘积为 c，其和为 b。例如，求解 x² + 7x + 12 = 0 时，我们找到乘积为 12 且和为 7 的两个数。这两个数是 3 和 4。

Thus, we write (x + 3)(x + 4) = 0. Setting each factor to zero gives x + 3 = 0 or x + 4 = 0, so x = −3 or x = −4.

因此，我们写成 (x + 3)(x + 4) = 0。令每个因式等于零，得到 x + 3 = 0 或 x + 4 = 0，所以 x = −3 或 x = −4。

When the coefficient of x² is not 1, factorisation becomes slightly more involved. We look for factors of a × c that add up to b, and then factor by grouping. For example, to solve 2x² + 7x + 3 = 0, we find factors of 2 × 3 = 6 that sum to 7. The factors are 1 and 6. We rewrite the middle term: 2x² + x + 6x + 3 = 0, then factor by grouping: x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0. Hence x = −1/2 or x = −3.

当 x² 的系数不为 1 时，因式分解会稍微复杂一些。我们寻找 a × c 的因数，使其和为 b，然后通过分组进行因式分解。例如，求解 2x² + 7x + 3 = 0 时，找到 2 × 3 = 6 中相加为 7 的因数。这两个因数是 1 和 6。我们重写中间项：2x² + x + 6x + 3 = 0，然后分组分解：x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0。因此 x = −1/2 或 x = −3。

4. Solving by Completing the Square | 配方法求解

Completing the square is a powerful algebraic technique that transforms a quadratic expression into a perfect square plus a constant. This method is especially useful when factorisation is not straightforward.

配方法是一种强大的代数技术，它可以将二次表达式变换为一个完全平方加上一个常数。当因式分解不简便时，这种方法尤其有用。

For the equation x² + 6x − 7 = 0, we first move the constant term to the right side: x² + 6x = 7. Next, we add the square of half the coefficient of x to both sides. Half of 6 is 3, and 3² = 9, so we add 9 to both sides: x² + 6x + 9 = 16.

对于方程 x² + 6x − 7 = 0，我们首先将常数项移到右边：x² + 6x = 7。接下来，我们在两边同时加上 x 系数一半的平方。6 的一半是 3，而 3² = 9，因此两边各加 9：x² + 6x + 9 = 16。

Now the left side is a perfect square: (x + 3)² = 16. Taking the square root of both sides gives x + 3 = ±4, so x = 1 or x = −7.

现在左边是一个完全平方：(x + 3)² = 16。对方程两边开平方，得到 x + 3 = ±4，所以 x = 1 或 x = −7。

For equations where a ≠ 1, we first divide both sides by a. For example, in 2x² − 8x + 5 = 0, we divide by 2 to get x² − 4x + 5/2 = 0. Then we proceed with completing the square.

对于 a ≠ 1 的方程，我们先将两边除以 a。例如，在 2x² − 8x + 5 = 0 中，我们将两边除以 2，得到 x² − 4x + 5/2 = 0。然后继续进行配方。

In general, completing the square for the expression x² + bx gives (x + b/2)² − (b/2)². This technique also helps in finding the turning point of a quadratic graph and in deriving the quadratic formula.

一般来说，对表达式 x² + bx 进行配方得到 (x + b/2)² − (b/2)²。这种技术也有助于寻找二次图像的顶点以及推导二次公式。

5. The Quadratic Formula | 二次公式法

The quadratic formula is a universal method for solving any quadratic equation. It is derived by completing the square on the general form ax² + bx + c = 0 and is stated below:

二次公式是一种通用的方法，可用于求解任何二次方程。它通过对一般形式 ax² + bx + c = 0 进行配方推导而来，公式如下：

x = (−b ± √(b² − 4ac)) / 2a

To use the formula, you simply substitute the values of a, b, and c from the standard form of the equation into the formula and simplify. For example, to solve 3x² − 5x − 2 = 0, we identify a = 3, b = −5, c = −2.

使用公式时，只需将方程标准形式中的 a、b、c 值代入公式并化简即可。例如，要求解 3x² − 5x − 2 = 0，我们识别出 a = 3，b = −5，c = −2。

Substituting these values gives x = (5 ± √(25 + 24)) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6. Therefore, x = 2 or x = −1/3.

代入这些值得到 x = (5 ± √(25 + 24)) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6。因此，x = 2 或 x = −1/3。

Note that the formula contains the discriminant b² − 4ac, which determines the nature of the roots. We will explore this concept in the next section.

注意，公式中包含判别式 b² − 4ac，它决定了根的性质。我们将在下一节探讨这一概念。

6. The Discriminant | 判别式

The discriminant, denoted by the symbol Δ (Greek capital letter delta), is the expression b² − 4ac inside the square root of the quadratic formula. It tells us about the number and type of roots without actually solving the equation.

判别式，用符号 Δ（希腊大写字母 delta）表示，是二次公式中被开方的部分 b² − 4ac。它告诉我们方程的根的数量和类型，而无需实际求解方程。

There are three possible cases for the discriminant:

判别式有三种可能的情况：

If b² − 4ac > 0, the equation has two distinct real roots.
如果 b² − 4ac > 0，方程有两个不相等的实数根。
If b² − 4ac = 0, the equation has exactly one repeated real root.
如果 b² − 4ac = 0，方程恰有一个重实数根。
If b² − 4ac < 0, the equation has no real roots (two complex roots).
如果 b² − 4ac < 0，方程没有实数根（有两个复数根）。

For example, the equation x² + 4x + 5 = 0 has a discriminant of 4² − 4(1)(5) = 16 − 20 = −4, which is negative. Hence, this equation has no real roots. Graphically, this means the parabola does not intersect the x-axis.

例如，方程 x² + 4x + 5 = 0 的判别式为 4² − 4(1)(5) = 16 − 20 = −4，是负数。因此，该方程没有实数根。从图形上看，这意味着抛物线不与 x 轴相交。

In IGCSE exams, the discriminant is often used to determine whether an equation can be solved, or to find the range of values for a parameter that gives a certain type of roots.

在 IGCSE 考试中，判别式常用于判断方程是否可以求解，或者求使根具有某种性质的参数的取值范围。

7. Solving Word Problems | 应用题求解

Quadratic equations frequently appear in real-world word problems. The key to solving these problems is to translate the given information into a quadratic equation correctly, solve it, and then interpret the solutions in the context of the problem.

二次方程经常出现在实际应用题中。解决这类问题的关键是将所给信息正确地转化为二次方程，求解该方程，然后在问题的情境中解释所得到的解。

Consider the following example: The length of a rectangle is 3 cm more than its width, and its area is 40 cm². Let the width be x cm. Then the length is (x + 3) cm. The area formula gives x(x + 3) = 40, which simplifies to x² + 3x − 40 = 0.

考虑以下示例：一个矩形的长比宽多 3 厘米，且其面积为 40 平方厘米。设宽为 x 厘米。则长为 (x + 3) 厘米。面积公式给出 x(x + 3) = 40，化简得 x² + 3x − 40 = 0。

Factorising gives (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a length cannot be negative, we discard x = −8. Therefore, the width is 5 cm and the length is 8 cm.

因式分解得到 (x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于长度不能为负，我们舍弃 x = −8。因此，宽为 5 厘米，长为 8 厘米。

When solving word problems, always check the reasonableness of your answers and ensure that any discarded solutions are explained.

在解决应用题时，务必检查答案的合理性，并解释被舍弃的解的原因。

8. 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 correspond to the x-intercepts of the parabola. The direction in which the parabola opens depends on the sign of a.

二次函数 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根对应于抛物线与 x 轴的交点。抛物线的开口方向取决于 a 的符号。

If a > 0, the parabola opens upwards, forming a U-shape with a minimum point. If a < 0, the parabola opens downwards, forming an inverted U-shape with a maximum point. The turning point of the parabola can be found by completing the square.

如果 a > 0，抛物线开口向上，形成 U 形并具有最小值点。如果 a < 0，抛物线开口向下，形成倒 U 形并具有最大值点。抛物线的顶点可通过配方法求得。

For example, the equation y = x² − 4x + 3 can be written as y = (x − 2)² − 1, so the turning point is at (2, −1). Its roots are x = 1 and x = 3, which are the x-intercepts of the graph.

例如，方程 y = x² − 4x + 3 可以写成 y = (x − 2)² − 1，因此顶点坐标为 (2, −1)。它的根是 x = 1 和 x = 3，即该图像与 x 轴的交点。

The discriminant also has a graphical meaning: if it is positive, the graph crosses the x-axis at two points; if it is zero, the graph touches the x-axis at one point; if it is negative, the graph does not touch the x-axis at all.

判别式也具有图形意义：如果它是正数，图像与 x 轴在两个点处相交；如果为零，图像与 x 轴在一个点处相切；如果为负数，图像完全不与 x 轴相交。

9. Common Mistakes to Avoid | 常见错误避免

Many students make avoidable errors when working with quadratic equations. One of the most common mistakes is forgetting to rearrange the equation into standard form before applying the quadratic formula. Always ensure that one side of the equation is zero.

许多学生在处理二次方程时会犯一些可避免的错误。最常见的错误之一是忘记在应用二次公式之前将方程整理为标准形式。务必确保方程的一边为零。

Another frequent error is sign mistakes when substituting negative values into the quadratic formula. For example, if b = −5, then −b = 5, which is easy to get wrong without careful attention.

另一个常见错误是在将负值代入二次公式时出现符号错误。例如，如果 b = −5，那么 −b = 5，如果不细心就很容易出错。

Additionally, students often forget that taking the square root of both sides in the completing the square method yields two solutions: one positive and one negative. Missing the ± sign leads to incomplete answers.

此外，学生经常忘记在配方法中对两边开平方会产生两个解：一个正和一个负。遗漏 ± 符号会导致答案不完整。

Finally, check whether the final answer makes sense in the original equation. Substituting your roots back into the original equation to verify them is a good habit and can help you catch errors.

最后，检查最终答案在原方程中是否合理。将所求得的根代回原方程进行验证是一个好习惯，可以帮助你发现错误。

10. Exam Tips and Practice | 考试技巧与练习

In the IGCSE exam, quadratic equations may appear in multiple-choice questions, short-answer questions, and extended problem-solving questions. Here are some practical tips to help you maximise your marks:

在 IGCSE 考试中，二次方程可能出现在选择题、简答题和拓展性问题中。以下是一些帮助你获取高分的实用技巧：

Always attempt factorisation first — it is usually faster. If you cannot factorise within a few seconds, switch to the quadratic formula or completing the square.
始终先尝试因式分解——它通常更快。如果几秒钟内无法分解，切换到二次公式或配方法。
Show all working clearly, especially when using the quadratic formula. Most marks in IGCSE are awarded for method, not just the final answer.
清晰展示所有步骤，尤其是在使用二次公式时。IGCSE 的大部分分数基于方法而非仅最终答案。
Redirect to standard form ax² + bx + c = 0 before solving.
在求解前将方程化为标准形式 ax² + bx + c = 0。
Read word problems carefully and define your variables clearly. Convert the information step by step.
仔细阅读应用题，明确定义变量。逐步转换所给信息。
Check your answers by substitution into both the original equation and the given problem context.
通过将答案代回原方程和应用情境中来检查答案。

Practice is essential. Solve as many past-paper questions as possible. Try to solve each problem using at least two different methods to deepen your understanding.

练习至关重要。尽可能多地解答真题。尝试用至少两种不同方法来求解每个问题，以加深理解。

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Solving Quadratic Equations | 二次方程求解