Topic 068: Solving Quadratic Equations | 主题068：解二次方程

📚 Topic 068: Solving Quadratic Equations | 主题068：解二次方程

Quadratic equations appear frequently in the Edexcel IGCSE Mathematics syllabus. In this revision topic we explore their definition, standard form, and the three main algebraic methods of solution: factorisation, the quadratic formula, and completing the square. We also look at the discriminant and the graphical meaning of roots.

二次方程在 Edexcel IGCSE 数学大纲中频繁出现。在本复习主题中，我们探讨其定义、标准形式以及三种主要的代数解法：因式分解法、公式法和配方法。我们还会学习判别式以及根的图象意义。

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 unknown variable is squared.

二次方程是次数为 2 的多项式方程。这意味着未知数的最高次数是平方。

The most general form is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

最一般的形式是 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。
The value of a cannot be zero, because if a = 0 the equation becomes linear, not quadratic.

a 的值不能为零，因为如果 a = 0，方程就变成一次方程，而不是二次方程。
Quadratic equations have at most two real solutions, also called roots.

二次方程最多有两个实数解，也称为根。

2. The Standard Form | 标准形式

Before solving a quadratic equation, it is usually helpful to rearrange it into standard form.

在解二次方程之前，通常需要将其整理成标准形式。

Standard form means one side of the equation is 0 and the other side is written as ax² + bx + c.

标准形式指的是等式一边为 0，另一边写成 ax² + bx + c。
For example, the equation x² = 5x − 6 can be rearranged to x² − 5x + 6 = 0.

例如，方程 x² = 5x − 6 可以整理为 x² − 5x + 6 = 0。
When solving with a calculator, always identify a, b and c correctly from the standard form.

使用计算器求解时，务必从标准形式中正确识别 a、b 和 c。

3. Solving by Factorising | 因式分解法

Factorising is often the quickest method when the quadratic expression has simple integer factors.

当二次表达式含有简单的整数因式时，因式分解法通常是最快的方法。

Step 1: Write the equation in standard form ax² + bx + c = 0.

第一步：将方程写成标准形式 ax² + bx + c = 0。

Step 2: Factorise the left side into two brackets.

第二步：将左边分解为两个括号相乘。

Step 3: Use the zero-product property: if A × B = 0, then A = 0 or B = 0.

第三步：使用零乘积性质：如果 A × B = 0，那么 A = 0 或 B = 0。

Example: Solve x² − 5x + 6 = 0. Since (x − 2)(x − 3) = 0, we get x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。因为 (x − 2)(x − 3) = 0，所以 x = 2 或 x = 3。
Check your answers by substituting them back into the original equation.

将答案代回原方程进行检验。

4. Solving by the Quadratic Formula | 公式法

The quadratic formula works for every quadratic equation, including those that cannot be factorised easily.

公式法适用于所有二次方程，包括那些不容易因式分解的方程。

x = (−b ± √(b² − 4ac)) / (2a)

This formula gives two solutions because of the ± sign: one uses the plus, the other uses the minus.

这个公式给出两个解，因为 ± 号：一个取加号，另一个取减号。
Always write the equation in standard form before substituting a, b and c.

在代入 a、b 和 c 之前，一定要先将方程写成标准形式。
Example: Solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2.

例：解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。

Substituting gives:

代入可得：

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

Therefore x = (−3 + 5)/4 = 0.5, or x = (−3 − 5)/4 = −2.

因此 x = (−3 + 5)/4 = 0.5，或 x = (−3 − 5)/4 = −2。

5. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This method is also useful for finding turning points of graphs.

配方法将二次式改写为 a(x + p)² + q 的形式。这个方法也常用于寻找图象的顶点。

As an example, solve x² + 6x + 5 = 0 by completing the square.

例如，用配方法解 x² + 6x + 5 = 0。
Take half of 6, which is 3, and write (x + 3)². Expanding gives x² + 6x + 9, so we adjust:

取 6 的一半，即 3，写成 (x + 3)²。展开得 x² + 6x + 9，因此我们需要调整：

x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4

So the equation becomes (x + 3)² − 4 = 0. Hence (x + 3)² = 4.

于是方程变为 (x + 3)² − 4 = 0。因此 (x + 3)² = 4。
Taking square roots gives x + 3 = ±2, so x = −1 or x = −5.

两边开平方得 x + 3 = ±2，所以 x = −1 或 x = −5。

6. Graphical Interpretation | 图象意义

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

二次方程的解是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。

If the graph crosses the x-axis at two distinct points, the equation has two distinct real roots.

如果图象与 x 轴有两个不同的交点，则方程有两个不同的实数根。
If the graph just touches the x-axis at one point, the equation has one repeated root.

如果图象与 x 轴相切于一点，则方程有一个重根。
If the graph does not touch the x-axis at all, the equation has no real roots.

如果图象与 x 轴没有交点，则方程没有实数根。

The sign of a determines the shape of the parabola: a > 0 gives a U-shape opening upwards, a < 0 gives an n-shape opening downwards.

a 的符号决定抛物线的形状：a > 0 时开口向上，a < 0 时开口向下。

7. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant, often written as Δ.

二次公式中 b² − 4ac 的表达式称为判别式，通常记作 Δ。

Discriminant 判别式	Nature of roots 根的性质
Δ > 0 Δ > 0	Two distinct real roots 两个不同的实数根
Δ = 0 Δ = 0	One repeated real root 一个实数重根
Δ < 0 Δ < 0	No real roots 无实数根

If Δ is a positive square number such as 4, 9 or 16, the roots are rational and the quadratic can usually be factorised.

如果 Δ 是正平方数，如 4、9 或 16，则根是有理数，通常可以通过因式分解求解。
If Δ > 0 but not a perfect square, the roots are irrational and come in conjugate pairs such as 1 ± √2.

如果 Δ > 0 但不是完全平方数，则根是无理数，且成共轭对出现，如 1 ± √2。

8. Word Problems and Modelling | 应用题与建模

Quadratic equations are often used to model real-life situations involving areas, projectile motion, and geometric problems.

二次方程常用于模拟涉及面积、抛体运动和几何问题的实际情境。

Worked example: A rectangle has length 4 cm longer than its width. Its area is 60 cm². Find its width.

例题：一个矩形的长比宽长 4 cm，面积为 60 cm²。求它的宽。

Let the width be w cm. Then the length is w + 4 cm.

设宽为 w cm，则长为 w + 4 cm。

w(w + 4) = 60 → w² + 4w − 60 = 0

Factorise: (w + 10)(w − 6) = 0, so w = −10 or w = 6.

因式分解：(w + 10)(w − 6) = 0，所以 w = −10 或 w = 6。
Since width cannot be negative, the valid answer is w = 6 cm.

因为宽度不能为负，所以有效答案是 w = 6 cm。

9. Common Mistakes and Revision Tips | 常见错误与复习建议

Avoid the following common pitfalls when solving quadratic equations.

解答二次方程时，要避免以下常见陷阱。

Mistake: Forgetting that a cannot be zero in ax² + bx + c = 0.

错误：忘记 ax² + bx + c = 0 中 a 不能为零。
Mistake: Sign errors when substituting negative numbers into the quadratic formula.

错误：将负数代入二次公式时出现符号错误。
Mistake: Splitting a product into factors before one side is zero.

错误：在一边化为零之前就把乘积分拆。
Mistake: Forgetting the ± sign when taking square roots.

错误：开平方后忘记 ± 号。
Tip: After finding roots, substitute them back to check for mistakes.

提示：求出根后，代回原方程进行验算。

For revision, practise all three algebraic methods and be ready to choose the most efficient one in the exam.

复习时，请练习所有三种代数方法，并学会在考试中选择最有效率的一种。

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Topic 068: Solving Quadratic Equations | 主题068：解二次方程