Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, geometry, and even real-life problem solving. In this article, we will explore what a quadratic equation is, the main methods for solving it, and some common pitfalls to avoid.

二次方程是IGCSE数学中最重要的主题之一。它们出现在代数、几何乃至实际生活问题中。在本文中，我们将探讨二次方程的定义、求解的主要方法以及一些常见的易错点。

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

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable (usually x) is 2. For example, x² − 4x + 3 = 0 is a quadratic equation because the highest exponent is 2.

二次方程是次数为2的多项式方程。这意味着变量（通常是x）的最高次幂是2。例如，x² − 4x + 3 = 0 是一个二次方程，因为最高指数是2。

A quadratic equation may have two real solutions, one real solution, or no real solutions, depending on its discriminant.

一个二次方程可能有两个实数解、一个实数解或没有实数解，这取决于它的判别式。

2. The General Form | 一般形式

The standard form of a quadratic equation is:

二次方程的标准形式为：

ax² + bx + c = 0

where a, b, and c are constants and a ≠ 0. For example, in 2x² − 3x + 1 = 0, we have a = 2, b = −3, and c = 1.

其中a、b和c是常数，且 a ≠ 0。例如，在 2x² − 3x + 1 = 0 中，a = 2, b = −3, c = 1。

If a = 0, the equation becomes linear, not quadratic. So the condition a ≠ 0 is essential.

如果 a = 0，方程将变成一次方程而不是二次方程。因此 a ≠ 0 是必要条件。

3. Method 1: Factorisation | 方法一：因式分解

Factorisation is often the quickest method when the quadratic has simple integer factors. The idea is to write the quadratic as a product of two linear expressions.

当二次式具有简单的整数因子时，因式分解通常是最快的方法。思路是把二次式写成两个一次表达式的乘积。

For example, solve x² + 5x + 6 = 0. We look for two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, since 2 × 3 = 6 and 2 + 3 = 5. Therefore:

例如，解 x² + 5x + 6 = 0。我们寻找两个数，它们相乘得6且相加得5。这两个数是2和3，因为 2 × 3 = 6 且 2 + 3 = 5。因此：

(x + 2)(x + 3) = 0

When the product of two expressions is zero, at least one of them must be zero. So x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

当两个表达式的乘积为零时，至少其中一个必须为零。所以 x + 2 = 0 或 x + 3 = 0，解得 x = −2 或 x = −3。

For quadratics where the coefficient of x² is not 1, look for factor pairs that work with the coefficients. For example, 2x² + 7x + 3 = 0 can be factorised as (2x + 1)(x + 3) = 0.

对于x²系数不为1的二次式，寻找与系数匹配的因式对。例如，2x² + 7x + 3 = 0 可以因式分解为 (2x + 1)(x + 3) = 0。

Check your factorisation by expanding the brackets.

通过展开括号来检查你的因式分解是否正确。
If the expression cannot be factorised with integers, use another method.

如果表达式无法用整数因式分解，请使用其他方法。

4. Method 2: Quadratic Formula | 方法二：求根公式

The quadratic formula works for every quadratic equation, whether or not it factorises easily. For ax² + bx + c = 0, the solutions are given by:

求根公式适用于所有二次方程，无论它是否能轻松因式分解。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) ÷ (2a)

Let’s solve 2x² − 4x − 6 = 0 using the formula. Here a = 2, b = −4, c = −6. Substitute:

让我们用公式解 2x² − 4x − 6 = 0。这里 a = 2, b = −4, c = −6。代入：

x = (4 ± √(16 + 48)) ÷ 4 = (4 ± √64) ÷ 4 = (4 ± 8) ÷ 4

So x = (4 + 8) ÷ 4 = 3, or x = (4 − 8) ÷ 4 = −1. The solutions are x = 3 and x = −1.

所以 x = (4 + 8) ÷ 4 = 3，或 x = (4 − 8) ÷ 4 = −1。解为 x = 3 和 x = −1。

The symbol ± means that you calculate both the sum and the difference of the two terms.

符号±表示你需要分别计算两项之和与差。

5. Method 3: Completing the Square | 方法三：配方法

Completing the square rewrites a quadratic expression in the form a(x − h)² + k. This is especially useful when solving equations and when working with parabolas.

配方法将二次表达式改写为 a(x − h)² + k 的形式。这在解方程和处理抛物线时尤其有用。

To complete the square for x² + 6x + 5 = 0, first take half of the coefficient of x (which is 6 ÷ 2 = 3) and square it (3² = 9). Rewrite the equation as:

要对 x² + 6x + 5 = 0 配方，先取x系数的一半（6 ÷ 2 = 3）并平方（3² = 9）。将方程重写为：

(x + 3)² − 9 + 5 = 0

Simplify: (x + 3)² − 4 = 0. Then (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.

化简：(x + 3)² − 4 = 0。然后 (x + 3)² = 4，所以 x + 3 = ±2，解得 x = −1 或 x = −5。

For quadratics where the coefficient a is not 1, first divide the whole equation by a before completing the square.

当a不等于1时，先对方程整体除以a，再进行配方。

6. Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root, b² − 4ac. It tells us how many real solutions a quadratic equation has.

判别式是求根公式中根号下的部分，即 b² − 4ac。它告诉我们一个二次方程有多少个实数解。

Value of b² − 4ac	Number of real solutions	判断
> 0	Two distinct real solutions	两个不同实数解
= 0	One repeated real solution	一个重根实数解
< 0	No real solutions	无实数解

For example, x² − 4x + 4 = 0 has discriminant 16 − 16 = 0, so it has exactly one solution, x = 2.

例如，x² − 4x + 4 = 0 的判别式为 16 − 16 = 0，所以它只有一个解，x = 2。

7. Solving Word Problems | 解应用题

Quadratic equations often arise from geometric or physical problems. A common example involves the area of a rectangle.

二次方程常常来自几何或物理问题。一个常见例子涉及矩形的面积。

Suppose a rectangle’s length is 3 metres longer than its width, and its area is 40 square metres. Let the width be x. Then the length is x + 3, and:

假设一个矩形的长比宽长3米，面积为40平方米。设宽为x，则长为 x + 3，于是：

x(x + 3) = 40 ⇒ x² + 3x − 40 = 0

Factorise: (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since a width cannot be negative, the width is 5 metres and the length is 8 metres.

因式分解：(x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于宽度不能为负数，宽度为5米，长度为8米。

Always check whether your solutions make sense in the context of the problem.

始终检查你的解在问题情境中是否有意义。

8. Graphical Interpretation | 图像意义

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

二次方程 y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解是抛物线与x轴交点的x坐标。

If the discriminant is positive, the parabola crosses the x-axis at two points. If it is zero, the parabola touches the x-axis at exactly one point. If it is negative, the parabola does not cross the x-axis at all.

如果判别式为正，抛物线与x轴有两个交点。如果为零，抛物线正好与x轴相切于一点。如果为负，抛物线完全不与x轴相交。

When a > 0, the parabola opens upwards; when a < 0, it opens downwards.

当 a > 0 时，抛物线开口向上；当 a < 0 时，开口向下。

9. Common Pitfalls | 常见错误

Students often make mistakes with signs, fractions, and forgetting to set the equation to zero. Here are some tips to avoid pitfalls.

学生常常在符号、分数以及忘记将方程设为零上犯错。以下是一些避免错误的建议。

Always rearrange the equation to the form ax² + bx + c = 0 before factorising or using the formula.

在因式分解或使用公式之前，务必把方程整理成 ax² + bx + c = 0 的形式。
Watch your signs when substituting into the quadratic formula. For example, if b = −5, then −b becomes 5.

代入求根公式时注意符号。例如，如果 b = −5，那么 −b 变成 5。
When solving by factorisation, do not divide both sides by a variable unless you are sure it is not zero.

用因式分解求解时，除非确定变量不为零，否则不要两边除以同一个变量。
Check your answers by substituting them back into the original equation.

将答案代回原方程进行检验。

10. Practice Questions | 练习题目

Try solving the following quadratic equations to test your understanding.

尝试解下列二次方程，检验你的理解。

x² − 7x + 12 = 0

x² − 7x + 12 = 0
2x² + 5x − 3 = 0

2x² + 5x − 3 = 0
x² + 4x − 1 = 0 (use the quadratic formula, leave your answer in surd form)

x² + 4x − 1 = 0 （使用求根公式，答案保留根号形式）
The sum of a number and its reciprocal is 3. Write an equation and solve it.

一个数与它的倒数之和为3。列出方程并求解。

Solutions: 1) x = 3 or 4. 2) x = 0.5 or −3. 3) x = −2 ± √5. 4) x = (3 ± √5) ÷ 2.

答案：1) x = 3 或 4。2) x = 0.5 或 −3。3) x = −2 ± √5。4) x = (3 ± √5) ÷ 2。

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Solving Quadratic Equations | 解一元二次方程