Solving Quadratic Equations by Factorisation | 通过因式分解解二次方程

📚 Solving Quadratic Equations by Factorisation | 通过因式分解解二次方程

Quadratic equations appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to real-life applications such as projectile motion. One of the most elegant and efficient methods for solving them is factorisation. In this revision article, you will learn the theory behind factorisation, step-by-step techniques, special cases, and common pitfalls, all with clear worked examples.

二次方程在 IGCSE 数学大纲中随处可见，从代数变形到实际应用（如抛体运动）都离不开它。求解二次方程最简洁有效的方法之一就是因式分解。在本文中，你将从理论入手，学习因式分解的分步技巧、特殊情况以及常见易错点，并配有清晰可懂的典型例题。

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

A quadratic equation is a polynomial equation of degree 2. The general form is \(ax^2+bx+c=0\), but in this article we avoid LaTeX and write the form using Unicode: ax² + bx + c = 0, where a, b and c are constants, and a ≠ 0. The term ax² gives the equation its characteristic curved graph, called a parabola.

二次方程是次数为 2 的多项式方程，其一般形式为 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。二次项 ax² 使方程具有其独特的曲线图像，称为抛物线。

For example, x² + 3x – 4 = 0 is a quadratic equation, while x² + 2x + 1 = 0 is another. When we solve a quadratic equation, we find the values of x that make the equation true. These values are called the roots, solutions, or zeros of the equation.

例如，x² + 3x – 4 = 0 是一个二次方程，而 x² + 2x + 1 = 0 也是。解二次方程，就是找出使方程成立的 x 的值，这些值称为方程的根、解或零点。

2. The Standard Form Needed for Factorisation | 因式分解所需的标准形式

Before factorising a quadratic equation, you must rearrange it so that the right-hand side is 0. The left-hand side should be written in the order ax² + bx + c. For instance, the equation x² = 5x – 6 must be rewritten as x² – 5x + 6 = 0 before factorisation.

在因式分解二次方程之前，必须将其化为右边为 0 的形式。左边应按 ax² + bx + c 的次序书写。例如，方程 x² = 5x – 6 必须改写为 x² – 5x + 6 = 0 再分解。

It is essential to remember that the coefficient of x² (the value of a) may be positive or negative. If a is negative, it is often helpful to multiply the whole equation by -1 so that a becomes positive, which simplifies factorisation.

务必记住，x² 的系数 a 可以是正数或负数。若 a 为负数，通常可将整个方程乘以 -1 使 a 变为正数，以简化因式分解。

Example: -x² + 4x – 3 = 0 can be multiplied by -1 to become x² – 4x + 3 = 0.

例如：-x² + 4x – 3 = 0 可两边乘以 -1，得到 x² – 4x + 3 = 0。

3. Why Does Factorising Work? | 为什么因式分解可行？

Factorising a quadratic expression means writing it as a product of two linear factors. The key idea comes from the zero product property: if the product of two expressions is zero, then at least one of the expressions must be zero. In symbols, if A × B = 0, then A = 0 or B = 0.

因式分解二次表达式，就是将其写成两个一次因式的乘积。其关键依据是零乘积性质：若两个表达式的乘积为零，则至少有一个表达式为零。用符号表示：若 A × B = 0，则 A = 0 或 B = 0。

For example, if (x – 2)(x + 3) = 0, then we know that either x – 2 = 0 or x + 3 = 0. Therefore x = 2 or x = -3. This is why factorising is such a powerful tool: it reduces the problem to two simple linear equations.

例如，若 (x – 2)(x + 3) = 0，那么必有 x – 2 = 0 或 x + 3 = 0。因此 x = 2 或 x = -3。这正是因式分解如此强大的原因：它将问题简化为两个简单的一次方程。

4. Basic Factorisation: Taking Out a Common Factor | 基本因式分解：提取公因子

The simplest form of factorisation is extracting a common factor from every term. This is often the first step in solving a quadratic equation, especially when the constant term c is zero.

最简单的因式分解是从每一项中提取公因子。这通常是解二次方程的第一步，尤其是当常数项 c 为零时。

Example: Solve 3x² + 9x = 0. Both terms have a common factor of 3x, so we write 3x(x + 3) = 0. By the zero product property, 3x = 0 or x + 3 = 0. Thus x = 0 or x = -3.

例：解方程 3x² + 9x = 0。各项含有公因子 3x，因此可写成 3x(x + 3) = 0。根据零乘积性质，3x = 0 或 x + 3 = 0，故 x = 0 或 x = -3。

Expression	Common Factor	Factorised Form
2x² + 6x	2x	2x(x + 3)
5x² – 10x	5x	5x(x – 2)
x² – 4x	x	x(x – 4)

Always check whether a common factor can be taken out first. Doing so makes the remaining factor smaller and easier to handle.

总是先检查能否提取公因子。这样能使剩余因子更简洁、更容易处理。

5. Factorising Quadratics with Leading Coefficient 1 | 首项系数为 1 的二次三项式因式分解

When a = 1, the quadratic has the form x² + bx + c. Our goal is to write it as (x + p)(x + q), where p and q are numbers. Expanding (x + p)(x + q) gives x² + (p + q)x + pq. Therefore, to factorise x² + bx + c, we must find two numbers p and q such that p + q = b and pq = c.

当 a = 1 时，二次式为 x² + bx + c。我们的目标是将其写成 (x + p)(x + q)，其中 p、q 为数值。展开 (x + p)(x + q) 得到 x² + (p + q)x + pq。因此，要将 x² + bx + c 因式分解，必须找到两个数 p、q，使得 p + q = b，且 pq = c。

Example: Factorise x² + 7x + 12. We need two numbers whose product is 12 and whose sum is 7. The numbers are 3 and 4, because 3 + 4 = 7 and 3 × 4 = 12. Hence x² + 7x + 12 = (x + 3)(x + 4).

例：分解 x² + 7x + 12。我们需要两个数，乘积为 12，和为 7。这两个数是 3 和 4，因为 3 + 4 = 7，3 × 4 = 12。因此 x² + 7x + 12 = (x + 3)(x + 4)。

If the constant term c is negative, the two required numbers will have opposite signs. For example, x² – 2x – 15: find two numbers with product -15 and sum -2. The numbers are -5 and +3, because (-5) + 3 = -2 and (-5) × 3 = -15. So x² – 2x – 15 = (x – 5)(x + 3).

如果常数项 c 为负数，所需两数符号相反。例如，x² – 2x – 15：找两个数，乘积为 -15，和为 -2。这两个数是 -5 和 +3，因为 (-5) + 3 = -2，(-5) × 3 = -15。因此 x² – 2x – 15 = (x – 5)(x + 3)。

6. Factorising Quadratics with Leading Coefficient Not 1 | 首项系数不为 1 的二次三项式因式分解

When a ≠ 1, factorising becomes more challenging. A common method is the ‘ac’ (or ‘product-sum’) method. For a quadratic ax² + bx + c, we find two numbers whose product is a × c and whose sum is b. Then we split the middle term and factor by grouping.

当 a ≠ 1 时，因式分解更具挑战性。常用方法是 ‘ac’（或乘积-和）法。对于二次式 ax² + bx + c，先找两个数，其乘积为 a × c，其和为 b。然后拆分中间项，并采用分组分解法。

Example: Factorise 2x² + 5x + 3. Here a = 2, b = 5, c = 3. The product a × c = 2 × 3 = 6. We need two numbers with product 6 and sum 5: these are 2 and 3. Split the middle term: 2x² + 2x + 3x + 3. Now group: (2x² + 2x) + (3x + 3) = 2x(x + 1) + 3(x + 1) = (x + 1)(2x + 3).

例：分解 2x² + 5x + 3。这里 a = 2，b = 5，c = 3。a × c = 2 × 3 = 6。需要两个数乘积为 6、和为 5：它们是 2 和 3。拆分中间项：2x² + 2x + 3x + 3。然后分组：(2x² + 2x) + (3x + 3) = 2x(x + 1) + 3(x + 1) = (x + 1)(2x + 3)。

This method works for any quadratic with integer coefficients that factorises neatly.

这种方法适用于所有能整齐因式分解的整数系数二次式。

7. Solving Quadratic Equations by Setting Factors to Zero | 通过令因子为零解二次方程

Once the quadratic is written as (x + p)(x + q) = 0, the solution follows immediately: either x + p = 0 or x + q = 0. Therefore x = -p or x = -q. Always give both solutions, often separated by ‘or’ or written as a set of values.

一旦将二次式写成 (x + p)(x + q) = 0 的形式，解立即可得：要么 x + p = 0，要么 x + q = 0。因此 x = -p 或 x = -q。务必写出两个解，通常用 ‘或’ 连接，或作为集合写出。

Worked example: Solve x² – 5x + 6 = 0. Factorise: (x – 2)(x – 3) = 0. So x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

例：解方程 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0。所以 x – 2 = 0 或 x – 3 = 0，解得 x = 2 或 x = 3。

Worked example: Solve 2x² + 5x + 3 = 0. From above, (x + 1)(2x + 3) = 0. Thus x + 1 = 0 or 2x + 3 = 0. So x = -1 or x = -3/2. Notice the fraction is written as -3/2 for clarity.

例：解方程 2x² + 5x + 3 = 0。如上面所得，(x + 1)(2x + 3) = 0。因此 x + 1 = 0 或 2x + 3 = 0。所以 x = -1 或 x = -3/2。这里分数写成 -3/2 以清晰表达。

8. The Difference of Two Squares | 平方差公式

A very important special pattern is the difference of two squares: x² – a² = (x – a)(x + a). This arises when the quadratic has no x-term (b = 0) and the constant term is a negative perfect square.

一个非常重要的特殊模式是平方差公式：x² – a² = (x – a)(x + a)。当二次式没有一次项（b = 0）且常数项为负的完全平方数时，就会使用这一公式。

Example: Solve x² – 9 = 0. Since 9 = 3², we write x² – 3² = 0, then (x – 3)(x + 3) = 0. Hence x = 3 or x = -3.

例：解方程 x² – 9 = 0。因为 9 = 3²，可写为 x² – 3² = 0，再得 (x – 3)(x + 3) = 0。因此 x = 3 或 x = -3。

This pattern also works when the coefficient a is itself a perfect square. For example, 4x² – 25 = (2x – 5)(2x + 5). Set each factor to zero: 2x – 5 = 0 → x = 5/2, and 2x + 5 = 0 → x = -5/2.

当 x² 的系数也是完全平方数时，该公式同样适用。例如，4x² – 25 = (2x – 5)(2x + 5)。令各因子为零：2x – 5 = 0，得 x = 5/2；2x + 5 = 0，得 x = -5/2。

9. Special Cases: Repeated Roots and Missing Terms | 特殊情况：重根与缺项

Some quadratic equations have only one unique solution, called a repeated root. For example, x² – 6x + 9 = 0 factorises as (x – 3)(x – 3) = 0, or (x – 3)² = 0. Setting the factor to zero gives x = 3. Even though the factor appears twice, the only solution is x = 3. In your answer, you may write ‘x = 3 (repeated)’ or simply x = 3.

有些二次方程只有一个解，称为重根。例如，x² – 6x + 9 = 0 可分解为 (x – 3)(x – 3) = 0，即 (x – 3)² = 0。令因子为零得 x = 3。尽管因子出现了两次，但唯一解仍是 x = 3。作答时，可以写 “x = 3（重根）”，或直接写 x = 3。

If the constant term is missing (c = 0), the equation always factorises by taking out x. For example, 4x² + 8x = 0 → 4x(x + 2) = 0, so x = 0 or x = -2. Note that one root is always x = 0.

如果常数项缺失（c = 0），方程总是可以通过提取 x 来因式分解。例如，4x² + 8x = 0 → 4x(x + 2) = 0，所以 x = 0 或 x = -2。注意其中一个根总是 x = 0。

10. Linking Factorisation to the Graph | 将因式分解与图像联系

Quadratic equations correspond to quadratic graphs. If a quadratic equation ax² + bx + c = 0 has roots r and s, then the graph of y = ax² + bx + c crosses or touches the x-axis at x = r and x = s. In other words, the roots are the x-intercepts of the parabola.

二次方程对应二次函数的图像。若二次方程 ax² + bx + c = 0 的根为 r 和 s，则函数 y = ax² + bx + c 的图像在 x = r 和 x = s 处与 x 轴相交或相切。换言之，方程的根就是抛物线与 x 轴交点的横坐标。

For example, y = x² – 5x + 6 = (x – 2)(x – 3) has x-intercepts at (2, 0) and (3, 0). If the equation has a repeated root, the graph touches the x-axis at one point, such as y = x² – 6x + 9 = (x – 3)² touching at (3, 0).

例如，y = x² – 5x + 6 = (x – 2)(x – 3) 与 x 轴交于点 (2, 0) 和 (3, 0)。若方程有重根，则图像与 x 轴相切于一点，例如 y = x² – 6x + 9 = (x – 3)² 在 (3, 0) 处相切。

This connection helps you check whether your factorised solutions are plausible. A positive value of a gives a U-shaped parabola; if a is negative, the parabola is inverted.

这种联系可帮助你检查分解出的解是否合理。a 为正时抛物线开口向上；a 为负时抛物线开口向下。

11. Common Mistakes and Top Tips | 常见易错点与高分技巧

Below are frequent errors students make, along with suggestions to avoid them.

以下是学生常犯的错误及避免这些错误的建议。

Forgetting to rearrange to 0: Always move all terms to one side before factorising.

忘记整理成 0： 分解前务必把所有项移到等号一侧。
Sign errors when choosing p and q: Double-check that p + q and pq match the coefficients.

选择 p、q 时出现符号错误： 再次检查 p + q 与 pq 是否与系数匹配。
Not taking out a common factor first: For example, 2x² + 8x + 6 should first be written as 2(x² + 4x + 3), then factored further.

没有先提取公因子： 例如 2x² + 8x + 6 应先写为 2(x² + 4x + 3)，再继续分解。
Confusing factorising with expanding: Factorising requires writing a product, not removing brackets.

混淆因式分解与展开： 因式分解是写成乘积形式，而不是去掉括号。
Missing the second root: Remember to solve each factor equal to zero, not just one.

遗漏第二个根： 记住要使每个因子分别等于零，而不只是其中一个。

Top tip: after finding your roots, substitute them back into the original equation to check your work.

高分技巧：求出根后，代回原方程即可验证结果是否正确。

12. Practice Questions | 练习与挑战

Try these problems on your own, then check the answers below.

请独立尝试以下题目，再核对下方的答案。

Solve x² + 5x + 6 = 0.

解方程 x² + 5x + 6 = 0。
Solve x² – 10x + 25 = 0.

解方程 x² – 10x + 25 = 0。
Solve 3x² – 12x = 0.

解方程 3x² – 12x = 0。
Solve 2x² – 7x + 3 = 0.

解方程 2x² – 7x + 3 = 0。
Solve 4x² – 1 = 0.

解方程 4x² – 1 = 0。

Answers: 1) x = -2 or x = -3. 2) x = 5 (repeated). 3) x = 0 or x = 4. 4) x = 1/2 or x = 3. 5) x = 1/2 or x = -1/2.

答案：1) x = -2 或 x = -3。2) x = 5（重根）。3) x = 0 或 x = 4。4) x = 1/2 或 x = 3。5) x = 1/2 或 x = -1/2。

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Solving Quadratic Equations by Factorisation | 通过因式分解解二次方程