Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in the Edexcel IGCSE Mathematics syllabus. In this article, we will explore the main methods for solving them, understand the meaning of the discriminant, and look at real-world applications.

二次方程在 Edexcel IGCSE 数学课程中频繁出现。在本文中，我们将探讨求解二次方程的主要方法，理解判别式的含义，并研究实际应用。

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.

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 的最高次数是 2。

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

如果 a = 0，方程就变成一次方程，而不是二次方程。
Every quadratic equation has exactly two solutions (roots), which may be real or complex, distinct or equal.

每个二次方程都有两个解（根），它们可能是实数或复数，可能不同或相等。
The standard form ax² + bx + c = 0 is the most convenient for solving.

标准形式 ax² + bx + c = 0 是求解最方便的形式。

2. Solving by Factorisation | 因式分解法

When the quadratic expression can be written as a product of two linear factors, we can solve by setting each factor equal to zero.

当二次表达式可以写成两个一次因式的乘积时，我们可以令每个因式等于零来求解。

Example: Solve x² – 5x + 6 = 0.

例如：解 x² – 5x + 6 = 0。

We factorise: (x – 2)(x – 3) = 0.

我们因式分解：(x – 2)(x – 3) = 0。

Therefore x = 2 or x = 3.

因此 x = 2 或 x = 3。

Steps for factorisation:

因式分解的步骤：

Write the equation in standard form ax² + bx + c = 0.

将方程写成标准形式 ax² + bx + c = 0。
Find two numbers that multiply to give ac and add to give b.

找到两个数，它们的乘积等于 ac，和等于 b。
Split the middle term and factor by grouping.

拆分中间项并分组因式分解。
Set each linear factor equal to zero and solve.

令每个一次因式等于零并求解。

If (px + q)(rx + s) = 0, then x = -q/p or x = -s/r.

如果 (px + q)(rx + s) = 0，那么 x = -q/p 或 x = -s/r。

3. Solving by Taking Square Roots | 平方根法

If the quadratic equation has no linear term, i.e. ax² + c = 0, we can solve directly by isolating x² and taking square roots.

如果二次方程没有一次项，即 ax² + c = 0，我们可以直接分离 x² 并取平方根来求解。

Example: Solve 2x² – 8 = 0.

例如：解 2x² – 8 = 0。

Add 8 to both sides: 2x² = 8.

两边加 8：2x² = 8。

Divide by 2: x² = 4.

两边除以 2：x² = 4。

Take square roots: x = ±2.

取平方根：x = ±2。

Remember that the square root of a positive number has two values: one positive and one negative.

记住正数的平方根有两个值：一正一负。

If x² = k (k ≥ 0), then x = ±√k.

如果 x² = k（k ≥ 0），那么 x = ±√k。

4. Completing the Square | 配方法

Completing the square is a powerful technique that rewrites ax² + bx + c in the form a(x – h)² + k. This is especially useful when factorisation is difficult or impossible.

配方法是一种强有力的技巧，可将 ax² + bx + c 改写为 a(x – h)² + k。这在因式分解困难或不可能时特别有用。

For a quadratic x² + bx + c, add and subtract (b/2)².

对于二次式 x² + bx + c，加上并减去 (b/2)²。

Example: Solve x² + 6x + 1 = 0 by completing the square.

例如：用配方法解 x² + 6x + 1 = 0。

Rewrite: x² + 6x = -1.

改写：x² + 6x = -1。

Add (6/2)² = 9 to both sides: x² + 6x + 9 = 8.

两边加上 (6/2)² = 9：x² + 6x + 9 = 8。

Now (x + 3)² = 8, so x + 3 = ±√8.

于是 (x + 3)² = 8，所以 x + 3 = ±√8。

Therefore x = -3 ± 2√2.

因此 x = -3 ± 2√2。

x² + bx + c = (x + b/2)² – (b/2)² + c

5. The Quadratic Formula | 二次求根公式

The quadratic formula solves any quadratic equation directly. It is derived from completing the square.

二次求根公式可以直接求解任何二次方程。它通过配方法推导得出。

For ax² + bx + c = 0, x = (-b ± √(b² – 4ac)) / (2a).

对于 ax² + bx + c = 0，x = (-b ± √(b² – 4ac)) / (2a)。

Example: Solve 2x² + 3x – 2 = 0.

例如：解 2x² + 3x – 2 = 0。

Here a = 2, b = 3, c = -2.

这里 a = 2，b = 3，c = -2。

Substitute into the formula:

代入公式：

x = (-3 ± √(3² – 4×2×(-2))) / (2×2) = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4.

x = (-3 ± √(3² – 4×2×(-2))) / (2×2) = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4。

So x = 2/4 = 1/2 or x = -8/4 = -2.

因此 x = 2/4 = 1/2 或 x = -8/4 = -2。

Always check your substitution signs carefully, especially when c is negative.

代入时务必小心符号，特别是当 c 为负数时。

6. The Discriminant | 判别式

The expression b² – 4ac inside the square root is called the discriminant, often denoted by Δ.

根号中的表达式 b² – 4ac 称为判别式，通常记作 Δ。

It tells us the nature of the roots without solving the equation.

它无需解方程就能告诉我们根的性质。

If Δ > 0, there are two distinct real roots.

如果 Δ > 0，有两个不同的实数根。
If Δ = 0, there is one repeated root (or two equal real roots).

如果 Δ = 0，有一个重根（或两个相等的实数根）。
If Δ < 0, there are no real roots; the roots are complex conjugates.

如果 Δ < 0，没有实数根；根是一对共轭复数。

Δ = b² – 4ac

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, so there is one repeated root.

Δ = (-4)² – 4×1×4 = 16 – 16 = 0，所以有一个重根。

7. Solving Word Problems with Quadratics | 二次方程应用题

Many word problems lead to quadratic equations. The key is to translate the problem into algebraic form and then solve.

许多应用题会导出二次方程。关键是将问题转化为代数形式，然后求解。

Example: The length of a rectangle is 3 cm longer than its width. The area is 40 cm². Find the width.

例如：一个长方形的长比宽长 3 cm，面积为 40 cm²。求宽。

Let the width be x cm. Then length = x + 3 cm.

设宽为 x cm，则长为 x + 3 cm。

Area equation: x(x + 3) = 40.

面积方程：x(x + 3) = 40。

Expanding: x² + 3x – 40 = 0.

展开：x² + 3x – 40 = 0。

Factorising: (x + 8)(x – 5) = 0.

因式分解：(x + 8)(x – 5) = 0。

So x = 5 or x = -8. Since width cannot be negative, x = 5 cm.

所以 x = 5 或 x = -8。由于宽不能为负，x = 5 cm。

Always reject solutions that do not make sense in the context of the problem.

始终舍去在问题情境中无意义的解。

8. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. The roots of the equation ax² + bx + c = 0 correspond to the x-intercepts of the graph.

y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的根对应于图像与 x 轴的交点。

If a > 0, the parabola opens upwards and has a minimum point.

如果 a > 0，抛物线开口向上，有最小值点。
If a < 0, the parabola opens downwards and has a maximum point.

如果 a < 0，抛物线开口向下，有最大值点。
The vertex lies on the axis of symmetry x = -b/(2a).

顶点位于对称轴 x = -b/(2a) 上。
The y-intercept is at (0, c).

y 截距为 (0, c)。

Using the discriminant, we know whether the parabola crosses the x-axis, touches it, or does not meet it at all.

利用判别式，我们可以知道抛物线是与 x 轴相交、相切还是完全不相交。

9. Common Mistakes to Avoid | 常见错误

Students often make errors when solving quadratics. Here are some important points to remember.

学生在求解二次方程时经常犯错。以下是一些需要记住的重要点。

Do not forget to set the equation to zero before factorising or using the formula.

在因式分解或使用公式之前，不要忘记将方程化为零。
When taking square roots, remember to include both ± signs if the value is positive.

取平方根时，若值为正，记得包含 ± 两个符号。
When using the quadratic formula, check that a, b and c are from the standard form ax² + bx + c = 0.

使用二次公式时，确认 a、b、c 来自标准形式 ax² + bx + c = 0。
Do not cancel terms incorrectly; divide both sides by a common factor instead.

不要错误地约分；应两边同时除以公因子。
In word problems, always interpret the solutions and reject invalid ones based on the context.

在应用题中，始终解释解的含义，并根据情境舍去无效解。

10. Practice Strategies | 练习策略

To master quadratic equations, practice a variety of questions. Start with factorisation, then complete the square, and finally use the formula.

要掌握二次方程，练习多种题型。从因式分解开始，然后配方法，最后使用公式。

Suggested practice plan:

建议练习计划：

Solve 5 equations by factorisation.

用因式分解法解 5 个方程。
Solve 5 equations by completing the square.

用配方法解 5 个方程。
Solve 5 equations using the quadratic formula.

用二次公式解 5 个方程。
Check the discriminant of 10 equations and classify the roots.

计算 10 个方程的判别式并分类根。
Attempt 5 word problems involving area, motion or number relationships.

尝试 5 个涉及面积、运动或数关系的应用题。

Remember to show all steps clearly to avoid careless errors.

记得清楚展示所有步骤，避免粗心错误。

Summary | 总结

Quadratic equations are a central topic in IGCSE Mathematics. You should be able to solve them by factorisation, completing the square, and the quadratic formula. The discriminant helps you determine the number and type of roots without solving fully. Regular practice will build confidence and accuracy.

二次方程是 IGCSE 数学的核心主题。你应该能够通过因式分解、配方法和二次公式来求解它们。判别式可以帮助你在不完全求解的情况下确定根的数量和类型。定期练习将建立信心并提高准确性。

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Solving Quadratic Equations | 解二次方程