Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most frequently tested topics in IGCSE Mathematics. It appears in almost every exam session, often across both Paper 2 and Paper 4, and it underpins many higher-level algebraic skills. In this guide, we will explore the standard form of quadratics, the main solving methods, the discriminant, graph sketching, and real-world applications — all tailored to what you actually need in the exam room.

二次方程是 IGCSE 数学中考查频率最高的知识点之一，几乎每次考试都会出现，通常同时涉及 Paper 2 和 Paper 4，并且是许多高阶代数技巧的基础。在本指南中，我们将系统讲解二次方程的标准形式、主要解法、判别式、函数图像以及实际应用——全部紧扣考试真题的考查要求。

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

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the unknown variable is 2. Its general form is written as:

二次方程是次数为 2 的多项式方程，意味着未知数的最高次数为 2。它的一般形式可以写成：

ax² + bx + c = 0, a ≠ 0

Here, x is the unknown, while a, b, and c are constants. The coefficient a cannot be zero, because if a = 0 then the equation reduces to a linear equation and has only one solution. Examples of quadratic equations include 2x² + 3x − 5 = 0, x² − 9 = 0, and 4x² = x + 1. Note that the last example is not in standard form yet, but it can be rearranged by bringing all terms to one side.

其中 x 是未知数，a、b、c 是常数。系数 a 不能为零，因为如果 a = 0，方程就退化成为一次方程，只能有一个解。二次方程的例子包括 2x² + 3x − 5 = 0、x² − 9 = 0 以及 4x² = x + 1。注意最后一个例子还不是标准形式，需要将所有项移到等号一侧进行整理。

2. Standard Form and Key Terms | 标准形式与关键术语

To solve a quadratic equation confidently, you must first be able to recognise its standard form. In standard form, the term ax² is always written first, followed by bx, then c, with the entire expression set equal to zero. The values of a, b, and c are simply the coefficients of the corresponding terms, and it is essential to read their signs correctly before applying any formula.

要想自信地求解二次方程，首先必须学会识别标准形式。在标准形式中，ax² 项总是写在最前面，其次为 bx 项，再是常数项 c，整个表达式等于零。a、b、c 就是对应项的系数，在套用任何公式之前，正确读出它们的正负号至关重要。

The solutions of a quadratic equation are called roots or zeros. These are the values of x that make the equation true. If you plot the graph of y = ax² + bx + c, the roots are exactly the x-coordinates where the curve crosses the x-axis. A quadratic equation can have two distinct roots, one repeated root, or no real roots at all — we will return to this idea in Section 6.

二次方程的解称为根（roots）或零点（zeros），它们是使等式成立的 x 值。如果画出 y = ax² + bx + c 的图像，根恰好就是曲线与 x 轴交点的横坐标。一个二次方程可以有两个不相等的根、一个重根，或者完全没有实数根——我们将在第 6 节中详细讨论这一点。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest solving method when it works, and it is the first technique you should try. The method relies on a simple logical rule called the zero product property:

因式分解法在适用时是最快捷的解法，也是你应当首先尝试的技巧。这个方法依赖于一个简单但重要的逻辑规则——零乘积性质：

If p × q = 0, then p = 0 or q = 0

若 p × q = 0，则 p = 0 或 q = 0

Consider the equation x² − 5x + 6 = 0. We need two numbers that multiply to give +6 and add to give −5. These numbers are −2 and −3. Therefore the equation factorises as (x − 2)(x − 3) = 0. By the zero product property, either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

我们来看方程 x² − 5x + 6 = 0。需要找到两个数，它们相乘等于 +6，相加等于 −5。这两个数就是 −2 和 −3。因此原方程可以分解为 (x − 2)(x − 3) = 0。根据零乘积性质，要么 x − 2 = 0，要么 x − 3 = 0，所以 x = 2 或 x = 3。

When the coefficient a is 1, the process is straightforward: find two integers whose product is c and whose sum is b. When a is not 1, you may need to use trial and error, grouping, or the “multiply ac” method. Another common case is the difference of two squares: x² − 9 = (x − 3)(x + 3) = 0, giving x = 3 or x = −3. The general solving procedure is:

当系数 a = 1 时，因式分解的过程非常直接：找到两个整数，它们的乘积为 c、和为 b。当 a ≠ 1 时，可能需要试算、分组分解，或使用”相乘 ac”的方法。另一个常见情形是平方差公式：x² − 9 = (x − 3)(x + 3) = 0，所以 x = 3 或 x = −3。一般的解题步骤如下：

Step 1: Rearrange to the standard form ax² + bx + c = 0. | 第 1 步：整理成标准形式 ax² + bx + c = 0。
Step 2: Factorise the left-hand side completely. | 第 2 步：将等号左边的二次式完全因式分解。
Step 3: Apply the zero product property to form two linear equations. | 第 3 步：运用零乘积性质得出两个一次方程。
Step 4: Solve each linear equation to obtain the roots. | 第 4 步：分别解两个一次方程，得到方程的根。

4. Solving by Completing the Square | 配方法

Completing the square is a powerful technique that rewrites a quadratic expression as a perfect square plus a constant. It is especially useful when the equation has irrational roots or when you later need to find the vertex of a parabola. The general idea is to transform x² + bx + c into the form (x + p)² + q.

配方法是一种重要的解题技巧，它把一个二次式改写为完全平方式加上一个常数的形式。当方程含有无理根或需要进一步求抛物线顶点时，配方法尤其有用。其核心思路是将 x² + bx + c 变形为 (x + p)² + q 的形式。

Let us solve x² + 6x + 2 = 0. First, take half of the coefficient of x, which is 6 ÷ 2 = 3, and square it to obtain 9. Then rewrite the equation as x² + 6x + 9 − 9 + 2 = 0, so (x + 3)² − 7 = 0. Now isolate the square:

我们来解 x² + 6x + 2 = 0。首先取 x 系数的 6 的一半，即 6 ÷ 2 = 3，然后平方得到 9。接下来把方程改写为 x² + 6x + 9 − 9 + 2 = 0，即 (x + 3)² − 7 = 0。现在分离出完全平方式：

(x + 3)² = 7 ⟹ x + 3 = ±√7 ⟹ x = −3 ± √7

The two solutions are x = −3 + √7 and x = −3 − √7. They are irrational, so factorisation would have been difficult here. In general, completing the square for x² + bx + c involves writing (x + b/2)² and then subtracting (b/2)². This method also reveals the turning point of the graph, as we will see in Section 8.

方程的两个根为 x = −3 + √7 和 x = −3 − √7。它们都是无理数，因此使用因式分解比较困难。一般情况下，对 x² + bx + c 配方需要写成 (x + b/2)²，再减去 (b/2)²。该方法还能直接揭示图像的转向点，我们将在第 8 节中看到。

5. The Quadratic Formula | 二次求根公式

The quadratic formula is the ultimate tool for solving any quadratic equation. It is derived from completing the square and works even when factorisation fails. If you memorise this one formula, you can solve any quadratic equation, no matter how messy the coefficients are.

二次求根公式是解任意二次方程的万能工具。它由配方法推导而来，即使在因式分解失败时也依然适用。只要你记住这个公式，无论系数多么复杂，都能解出任何二次方程。

x = [−b ± √(b² − 4ac)] ÷ (2a)

To use this formula, first write the equation in standard form and identify a, b, and c. Then substitute them into the formula carefully. For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4, and c = −3. Substituting gives:

使用该公式时，先将方程写成标准

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Solving Quadratic Equations | 解二次方程