Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, geometry, and even in real-world problems. In this revision guide, we will explore the standard form, methods of solution, the discriminant, and common pitfalls, with clear worked examples.

二次方程是 IGCSE 数学中最重要的内容之一。它出现在代数、几何甚至现实生活问题中。在本复习指南中，我们将探讨标准形式、求解方法、判别式以及常见陷阱，并配有清晰的例题。

1. Standard Form | 标准形式

A quadratic equation is an equation that can be written in the form:

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. If a = 0, the equation becomes linear, not quadratic.

二次方程是能写成以下形式的方程：

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。如果 a = 0，方程就变成一次方程，而不是二次方程。

For example, 2x² + 3x – 5 = 0 is a quadratic equation. Make sure you can identify the values of a, b, and c.

例如，2x² + 3x – 5 = 0 是一个二次方程。确保你能识别出 a、b、c 的值。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic expression can be factored into two linear factors. The idea is to rewrite ax² + bx + c as (px + q)(rx + s), then set each factor equal to zero.

因式分解法在二次表达式能分解为两个一次因式时往往是最快的方法。其思路是将 ax² + bx + c 改写成 (px + q)(rx + s)，然后令每个因式等于零。

Worked example: Solve x² – 5x + 6 = 0.

例题：解 x² – 5x + 6 = 0。

(x – 2)(x – 3) = 0

Therefore x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

因此 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

Always check your factors by expanding them back to the original expression.

务必通过展开因式来检查是否与原表达式一致。

3. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that rewrites a quadratic expression in the form (x + p)² + q. This method is essential for deriving the quadratic formula and for finding the vertex of a parabola.

配方是一种强大的技巧，它将二次表达式改写为 (x + p)² + q 的形式。这种方法对于推导求根公式以及寻找抛物线的顶点至关重要。

Worked example: Solve x² + 6x – 4 = 0 by completing the square.

例题：用配方法解 x² + 6x – 4 = 0。

First, move the constant term to the other side: x² + 6x = 4.

首先，将常数项移到另一边：x² + 6x = 4。

Take half of 6, square it, and add to both sides: (6/2)² = 9, so x² + 6x + 9 = 4 + 9.

取 6 的一半，平方后加到两边：(6/2)² = 9，所以 x² + 6x + 9 = 4 + 9。

(x + 3)² = 13

Then x + 3 = ±√13, so x = –3 ± √13.

然后 x + 3 = ±√13，所以 x = –3 ± √13。

4. Solving by the Quadratic Formula | 公式法

The quadratic formula can solve any quadratic equation, even when factorisation is difficult or impossible. It is derived from completing the square and is given by:

求根公式可以解出任何二次方程，即使因式分解困难或不可能时也有效。它由配方法推导而来，公式为：

x = (–b ± √(b² – 4ac)) / (2a)

Worked example: Solve 2x² – 3x – 2 = 0 using the formula. Here a = 2, b = –3, c = –2.

例题：使用公式法解 2x² – 3x – 2 = 0。这里 a = 2，b = –3，c = –2。

x = (3 ± √(9 + 16)) / 4 = (3 ± 5) / 4

So x = (3 + 5)/4 = 2 or x = (3 – 5)/4 = –1/2.

因此 x = (3 + 5)/4 = 2 或 x = (3 – 5)/4 = –1/2。

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.

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

If b² – 4ac > 0, there are two distinct real roots.

若 b² – 4ac > 0，则有两个不同的实数根。
If b² – 4ac = 0, there is exactly one repeated real root.

若 b² – 4ac = 0，则有一个二重实数根。
If b² – 4ac < 0, there are no real roots (the roots are complex).

若 b² – 4ac < 0，则没有实数根（根为复数）。

For example, x² + 2x + 5 = 0 has discriminant 4 – 20 = –16, so it has no real solutions.

例如，x² + 2x + 5 = 0 的判别式为 4 – 20 = –16，因此它没有实数解。

6. Roots and Coefficients | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum and product of the roots can be found directly from the coefficients:

对于二次方程 ax² + bx + c = 0，其根为 α 和 β，根的和与积可以直接由系数求得：

α + β = –b/a, αβ = c/a

This is very useful for forming a quadratic equation when given its roots.

当给定根时，这个关系对于构造二次方程非常有用。

Worked example: Find a quadratic equation with roots 3 and –2. Sum = 1, product = –6, so the equation is x² – (sum)x + product = 0, i.e. x² – x – 6 = 0.

例题：求以 3 和 –2 为根的二次方程。和 = 1，积 = –6，所以方程为 x² – (和)x + 积 = 0，即 x² – x – 6 = 0。

7. Quadratic Graphs | 二次函数图像

The graph of a quadratic function y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards; if a < 0, it opens downwards. The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph.

二次函数 y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，开口向下。方程 ax² + bx + c = 0 的根就是图像与 x 轴交点的横坐标。

The vertex of the parabola has x-coordinate given by –b/(2a), and the y-coordinate can be found by substituting this value into the equation.

抛物线的顶点横坐标为 –b/(2a)，将其代入方程即可求得纵坐标。

For example, y = x² – 4x + 3 has vertex at x = 2, y = –1. Its roots are x = 1 and x = 3.

例如，y = x² – 4x + 3 的顶点在 x = 2，y = –1。其根为 x = 1 和 x = 3。

8. Applications | 实际应用

Quadratic equations are used to model projectile motion, area problems, and economic profit functions. For example, the height of a ball thrown upwards can be described by h(t) = –5t² + 20t + 1.

二次方程用于模拟抛体运动、面积问题以及经济利润函数。例如，向上抛出的球的高度可以用 h(t) = –5t² + 20t + 1 来描述。

To find when the ball hits the ground, set h(t) = 0 and solve for t. Only positive values of t are meaningful in most real-life contexts.

要求球何时落地，设 h(t) = 0 并解出 t。在大多数现实情境中，只有正的 t 值有意义。

When solving word problems, carefully define your variables, translate the information into an equation, and check that your answers make sense in the original context.

在解决文字应用题时，要仔细定义变量，将信息转化为方程，并检查答案在原始情境中是否合理。

9. Common Mistakes | 常见错误

Students often forget to rearrange the equation into the standard form ax² + bx + c = 0 before solving. For example, x² = 5x – 6 must be rewritten as x² – 5x + 6 = 0.

学生在求解前常忘记将方程整理为标准形式 ax² + bx + c = 0。例如，x² = 5x – 6 必须改写为 x² – 5x + 6 = 0。

Another common mistake is dividing both sides by x when x might be zero, losing solutions. Never divide by a variable term unless you are sure it cannot be zero.

另一个常见错误是当 x 可能为零时，两边同时除以 x，导致失根。绝不要除以含有变量的项，除非你确定它不可能为零。

Always expand and simplify before factoring.

因式分解前务必先展开并化简。
Use brackets correctly when substituting negative numbers into the quadratic formula.

将负数代入求根公式时，正确使用括号。
Check your solutions by substituting them back into the original equation.

将解代回原方程进行验证。

10. Practice Questions | 练习

Try these typical IGCSE questions. Solve each equation and write down the discriminant first where useful.

尝试以下典型 IGCSE 题目。先写出判别式（如适用），再解方程。

1. x² – 7x + 12 = 0	2. 2x² + 5x – 3 = 0
3. x² + 4x + 1 = 0 (give answers in surd form)	4. (x – 3)² = 25

Answers: 1) x = 3 or 4. 2) x = 0.5 or –3. 3) x = –2 ± √3. 4) x = 8 or –2.

答案：1) x = 3 或 4。2) x = 0.5 或 –3。3) x = –2 ± √3。4) x = 8 或 –2。

11. Summary | 总结

Quadratic equations can be solved by factorisation, completing the square, or using the quadratic formula. The discriminant tells us the nature of the roots. Understanding how to manipulate these equations is essential for IGCSE Mathematics.

二次方程可以通过因式分解、配方或求根公式来求解。判别式告诉我们根的性质。理解如何操作这些方程对于 IGCSE 数学至关重要。

Memorise the standard form, the quadratic formula, and the discriminant. Practise many problems until these steps become automatic.

记住标准形式、求根公式和判别式。多练习题目，直到这些步骤变得熟练。

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Solving Quadratic Equations | 解二次方程