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 IGCSE Mathematics. It appears in algebra, graphs, geometry and problem-solving questions. Understanding how to solve quadratic equations gives you a strong foundation for higher-level maths.

二次方程是IGCSE数学中最重要的主题之一。它出现在代数、图像、几何和应用题中。掌握解二次方程的方法，能为你学习更高级的数学打下坚实基础。

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

A quadratic equation is an equation where the highest power of the variable is 2. In its simplest form, it can be written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The term ax² is called the quadratic term, bx is the linear term, and c is the constant term.

二次方程是变量最高次数为2的方程。其最简形式可写为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。ax² 称为二次项，bx 称为一次项，c 称为常数项。

For example, 2x² – 5x + 3 = 0 is a quadratic equation. However, x² = 4 and (x – 1)(x + 2) = 0 are also quadratic equations because they can be expanded or rearranged into the standard form.

例如，2x² – 5x + 3 = 0 是一个二次方程。同样，x² = 4 和 (x – 1)(x + 2) = 0 也是二次方程，因为它们可以展开或整理成标准形式。

2. Standard Form and Key Features | 标准形式与关键特征

Before solving, always rearrange the equation into the standard form ax² + bx + c = 0. This means moving all terms to one side, leaving 0 on the other side. For example, x² = 6x – 8 becomes x² – 6x + 8 = 0.

在求解之前，务必先将方程整理成标准形式 ax² + bx + c = 0。这意味着把所有项移到一边，使另一边为0。例如，x² = 6x – 8 应化为 x² – 6x + 8 = 0。

The graph of a quadratic equation y = ax² + bx + c is a curve called a parabola. If a > 0, the parabola opens upwards like a ‘U’. If a < 0, it opens downwards like an 'n'. The solutions of the equation are the x-coordinates where the graph crosses the x-axis.

二次方程 y = ax² + bx + c 的图像是一条称为抛物线的曲线。如果 a > 0，抛物线开口向上，形如 ‘U’；如果 a < 0，抛物线开口向下，形如 'n'。方程的解就是图像与 x 轴交点的 x 坐标。

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the equation has simple integer factors. To solve x² + bx + c = 0, find two numbers that multiply to give c and add to give b.

当方程具有简单的整数因子时，因式分解通常是最快的方法。要解 x² + bx + c = 0，需要找到两个数，它们的乘积等于 c，和等于 b。

Example: Solve x² – 5x + 6 = 0. The two numbers are -2 and -3 because (-2) × (-3) = 6 and (-2) + (-3) = -5. Therefore (x – 2)(x – 3) = 0, so x = 2 or x = 3.

示例：解 x² – 5x + 6 = 0。两个数是 -2 和 -3，因为 (-2) × (-3) = 6，且 (-2) + (-3) = -5。因此 (x – 2)(x – 3) = 0，所以 x = 2 或 x = 3。

For equations with a leading coefficient not equal to 1, such as 2x² – 7x + 3 = 0, you may need to split the middle term or use trial and error. Here, (2x – 1)(x – 3) = 0 gives x = ½ or x = 3.

对于首项系数不为1的方程，如 2x² – 7x + 3 = 0，可能需要拆项或试错。这里 (2x – 1)(x – 3) = 0，得到 x = ½ 或 x = 3。

4. Solving by Completing the Square | 配方法

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

配方法将二次式改写成 a(x – p)² + q 的形式。这有助于找到顶点坐标，也用于解不易因式分解的方程。

Example: Solve x² + 6x – 2 = 0 by completing the square. First, x² + 6x = (x + 3)² – 9. So the equation becomes (x + 3)² – 9 – 2 = 0, that is (x + 3)² = 11. Taking square roots gives x + 3 = ±√11, so x = -3 ± √11.

示例：用配方法解 x² + 6x – 2 = 0。首先，x² + 6x = (x + 3)² – 9。于是方程变为 (x + 3)² – 9 – 2 = 0，即 (x + 3)² = 11。两边开平方得 x + 3 = ±√11，所以 x = -3 ± √11。

When a ≠ 1, factor out a first. For example, 2x² – 8x + 1 = 0 can be written as 2(x² – 4x) + 1 = 0, then complete the square inside the bracket.

当 a ≠ 1 时，先提出 a。例如，2x² – 8x + 1 = 0 可写成 2(x² – 4x) + 1 = 0，然后在括号内配方。

5. The Quadratic Formula | 二次求根公式

The quadratic formula works for all quadratic equations, including those that cannot be factorised. The formula gives the solutions of ax² + bx + c = 0 as follows:

二次求根公式适用于所有二次方程，包括无法因式分解的方程。对于 ax² + bx + c = 0，求根公式如下：

x = (-b ± √(b² – 4ac)) ÷ 2a

To use the formula, substitute the values of a, b and c carefully. Be sure to write the equation in standard form first.

使用公式时，要仔细代入 a、b、c 的值。务必先将方程写成标准形式。

Example: Solve x² + 2x – 5 = 0 using the formula. Here a = 1, b = 2, c = -5. Then b² – 4ac = 4 + 20 = 24, so x = (-2 ± √24) ÷ 2 = (-2 ± 2√6) ÷ 2 = -1 ± √6.

示例：用公式解 x² + 2x – 5 = 0。这里 a = 1，b = 2，c = -5。则 b² – 4ac = 4 + 20 = 24，所以 x = (-2 ± √24) ÷ 2 = (-2 ± 2√6) ÷ 2 = -1 ± √6。

6. 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.

二次求根公式中的 b² – 4ac 称为判别式。它告诉我们二次方程有多少个实数根。

If b² – 4ac > 0, the equation has two distinct real roots.
If b² – 4ac = 0, the equation has exactly one real root (two equal roots).
If b² – 4ac < 0, the equation has no real roots.

判别式大于0时，方程有两个不同的实数根；等于0时，方程有一个实数根（两个相等的根）；小于0时，方程没有实数根。

For example, the equation 2x² – 4x + 2 = 0 has discriminant (-4)² – 4(2)(2) = 16 – 16 = 0, so it has one repeated root x = 1.

例如，方程 2x² – 4x + 2 = 0 的判别式为 (-4)² – 4(2)(2) = 16 – 16 = 0，所以它有一个重根 x = 1。

7. Roots and the x-axis | 根与 x 轴

The real roots of a quadratic equation are the x-intercepts of its graph. A positive discriminant means the parabola crosses the x-axis twice. A zero discriminant means the parabola just touches the x-axis. A negative discriminant means the parabola never touches the x-axis.

二次方程的实数根就是其图像与 x 轴的交点。判别式为正时，抛物线与 x 轴相交两次；判别式为零时，抛物线恰好与 x 轴相切；判别式为负时，抛物线不与 x 轴相交。

The turning point of a parabola is found using x = -b ÷ 2a. Substituting this x-value into the equation gives the y-coordinate of the vertex. For example, y = x² – 4x + 3 has vertex at x = 2, y = -1.

抛物线的顶点横坐标由 x = -b ÷ 2a 求得。将这个 x 值代入方程，即可得到顶点的 y 坐标。例如，y = x² – 4x + 3 的顶点为 x = 2 时，y = -1。

8. Sum and Product of Roots | 根的和与积

If the roots of ax² + bx + c = 0 are α and β, then the sum of the roots is α + β = -b/a, and the product of the roots is αβ = c/a. These relationships are called Vieta’s formulas.

若 ax² + bx + c = 0 的两个根为 α 和 β，则根的和为 α + β = -b/a，根的积为 αβ = c/a。这些关系称为韦达定理。

These formulas are useful for checking solutions or for constructing a quadratic equation from its roots. For instance, if the roots are 2 and 5, then the equation can be written as x² – (2 + 5)x + (2 × 5) = 0, which is x² – 7x + 10 = 0.

这些公式可用于检验解，或根据根来构造二次方程。例如，若根为 2 和 5，则方程可写为 x² – (2 + 5)x + (2 × 5) = 0，即 x² – 7x + 10 = 0。

9. Problem Solving with Quadratics | 二次方程应用题

Many real-world problems lead to quadratic equations. For example, the area of a rectangle is given by length × width. If a rectangle has a length of (x + 5) cm and a width of (x – 2) cm, and the area is 24 cm², then:

许多实际问题会导出二次方程。例如，矩形的面积等于长乘以宽。若一个矩形的长为 (x + 5) cm，宽为 (x – 2) cm，面积为 24 cm²，则：

(x + 5)(x – 2) = 24 → x² + 3x – 10 = 24 → x² + 3x – 34 = 0

Solving this gives x = (-3 ± √145) ÷ 2. Only the positive value is valid because lengths cannot be negative.

解这个方程得 x = (-3 ± √145) ÷ 2。由于长度不能为负，只取正数解。

When solving word problems, define your variable, form the quadratic equation, solve it, and then check which solutions make sense in the context.

解应用题时，先设变量，再列出二次方程，求解后检查哪些解在题目情境中合理。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

One common mistake is forgetting to put the equation into standard form before using the quadratic formula. Always rearrange to ax² + bx + c = 0 first.

一个常见错误是使用求根公式前忘记将方程化为标准形式。务必先整理成 ax² + bx + c = 0。

Another mistake is dropping the ± sign when taking square roots. Remember that x² = k has two solutions: x = √k and x = -√k.

另一个错误是开平方时漏掉 ± 号。记住 x² = k 有两个解：x = √k 和 x = -√k。

Always show your working clearly, especially when factorising or substituting into the formula.
Check your answers by substituting them back into the original equation.
If the question asks for answers to a certain degree of accuracy, round only at the final step.

务必清晰展示解题步骤，尤其是因式分解或代入公式的过程。将解代回原方程检验答案。如果题目要求保留一定精度，只在最后一步四舍五入。

11. Practice Questions | 练习题

Try these questions to test your understanding. Answers are given after each question.

尝试以下题目来检验你的理解。每题后附有答案。

Solve by factorisation: x² – 3x – 10 = 0. Answer: x = 5 or x = -2.
Solve by completing the square: x² – 6x + 4 = 0. Answer: x = 3 ± √5.
Use the quadratic formula: 2x² + 5x – 3 = 0. Answer: x = ½ or x = -3.
Find the discriminant of x² + 3x + 8 = 0 and state the number of real roots. Answer: 9 – 32 = -23, so no real roots.
Write a quadratic equation with roots 4 and -1. Answer: x² – 3x – 4 = 0.

第1题答案：x = 5 或 x = -2。第2题答案：x = 3 ± √5。第3题答案：x = ½ 或 x = -3。第4题：判别式 9 – 32 = -23，无实数根。第5题答案：x² – 3x – 4 = 0。

12. Summary | 总结

A quadratic equation can be solved by factorisation, completing the square or using the quadratic formula. The discriminant b² – 4ac determines the number of real roots. Always check your answers and understand which method is most suitable for each problem.

二次方程可以通过因式分解法、配方法或求根公式来求解。判别式 b² – 4ac 决定了实数根的个数。始终检验答案，并判断每种问题最适合哪种方法。

With regular practice, solving quadratic equations becomes quick and accurate. Master this topic and you will find many other areas of IGCSE Mathematics much easier.

通过定期练习，解二次方程会变得又快又准。掌握这个主题，你会发现IGCSE数学中的许多其他领域也容易得多。

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Solving Quadratic Equations | 解二次方程