Solving Quadratic Equations by Factorisation | 因式分解求解二次方程

📚 Solving Quadratic Equations by Factorisation | 因式分解求解二次方程

Quadratic equations appear frequently in IGCSE Mathematics, and solving them by factorisation is one of the most direct methods. This article explains the steps clearly, with worked examples and common pitfalls to help you master the technique.

二次方程在 IGCSE 数学中频繁出现，而因式分解法是求解它们最直接的方法之一。本文将清晰讲解步骤，配合例题和常见易错点，帮助你掌握这一技巧。

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

A quadratic equation is an equation of degree 2, meaning the highest power of the variable is 2. In its simplest form, it can be written as \( ax^2 + bx + c = 0 \), where \( a \neq 0 \), and \( a \), \( b \), \( c \) are constants. The graph of a quadratic is a parabola.

二次方程是次数为 2 的方程，即变量的最高次数为 2。其最简形式可写成 ax² + bx + c = 0，其中 a ≠ 0，a、b、c 为常数。二次方程的图像是一条抛物线。

Examples of quadratic equations include: \( x^2 – 5x + 6 = 0 \), \( 2x^2 + 3x – 2 = 0 \), and \( x^2 = 9 \). Not all quadratic equations can be solved easily by factorisation, but many IGCSE questions are designed to be factorisable.

二次方程的例子包括：x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 x² = 9。并非所有二次方程都能轻松因式分解，但许多 IGCSE 题目都设计成可以分解的。

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2. Standard Form of a Quadratic Equation | 二次方程的标准形式

Before solving, it is essential to rearrange the equation into standard form: everything on one side, set equal to zero. The standard form is:

求解前，必须将方程整理为标准形式：所有项移到等号一边，使另一边为零。标准形式为：

ax² + bx + c = 0, a ≠ 0

For example, the equation \( x^2 = 6 – x \) must be rewritten as \( x^2 + x – 6 = 0 \). Only when one side is zero can we use the zero product property.

例如，方程 x² = 6 − x 必须改写为 x² + x − 6 = 0。只有一边为零时，才能使用零乘积性质。

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3. The Zero Product Property | 零乘积性质

The key idea behind solving by factorisation is: if the product of two expressions is zero, then at least one of the expressions must be zero. In symbols:

因式分解求解的关键思路是：如果两个表达式的乘积为零，那么至少其中一个表达式必须为零。用符号表示：

If A × B = 0, then A = 0 or B = 0

Once a quadratic is factored into the form \((x + p)(x + q) = 0\), we can split it into two linear equations: \(x + p = 0\) or \(x + q = 0\). Solving these gives the two roots of the original equation.

一旦二次方程被分解成 (x + p)(x + q) = 0 的形式，我们就能拆成两个一次方程：x + p = 0 或 x + q = 0。解这两个方程便得到原方程的两个根。

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4. Factorising Quadratics with a = 1 | 分解二次项系数为 1 的二次式

When \(a = 1\), the quadratic has the form \(x^2 + bx + c\). We look for two numbers that multiply to \(c\) and add to \(b\). For example, to factorise \(x^2 – 5x + 6\), we need two numbers whose product is 6 and sum is −5. These are −2 and −3.

当 a = 1 时，二次式为 x² + bx + c 的形式。我们寻找两个数，它们相乘等于 c，相加等于 b。例如，分解 x² − 5x + 6，需要两个数乘积为 6，和为 −5。它们是 −2 和 −3。

Thus \(x^2 – 5x + 6 = (x – 2)(x – 3)\). Always check by expanding: \((x-2)(x-3) = x^2 – 3x – 2x + 6 = x^2 – 5x + 6\).

因此 x² − 5x + 6 = (x − 2)(x − 3)。始终通过展开检查：(x − 2)(x − 3) = x² − 3x − 2x + 6 = x² − 5x + 6。

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5. Factorising Quadratics with a ≠ 1 | 分解二次项系数不为 1 的二次式

When \(a \neq 1\), such as \(2x^2 + 7x + 3\), we can use the method of splitting the middle term. Multiply \(a\) and \(c\): \(2 \times 3 = 6\). Find two numbers whose product is 6 and sum is 7: these are 1 and 6.

当 a ≠ 1 时，例如 2x² + 7x + 3，我们可以使用拆中项法。将 a 与 c 相乘：2 × 3 = 6。找两个数，乘积为 6，和为 7：它们是 1 和 6。

Rewrite the middle term \(7x\) as \(1x + 6x\), then factor by grouping:

将中间项 7x 改写成 1x + 6x，然后分组分解：

2x² + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)

So \(2x^2 + 7x + 3 = (2x + 1)(x + 3)\). Practice this method until it feels natural.

所以 2x² + 7x + 3 = (2x + 1)(x + 3)。练习这种方法直到熟练。

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6. Solving by Factorisation: Worked Example | 用因式分解求解：例题演示

Solve \(x^2 – 3x – 10 = 0\). Step 1: Factorise. Two numbers that multiply to −10 and add to −3 are −5 and 2. So \((x – 5)(x + 2) = 0\).

解方程 x² − 3x − 10 = 0。第一步：因式分解。两个数相乘为 −10，相加为 −3，它们是 −5 和 2。所以 (x − 5)(x + 2) = 0。

Step 2: Apply zero product property. Either \(x – 5 = 0\) or \(x + 2 = 0\). Therefore \(x = 5\) or \(x = -2\).

第二步：应用零乘积性质。要么 x − 5 = 0，要么 x + 2 = 0。因此 x = 5 或 x = −2。

Step 3: Write the solution set clearly: \(x = 5\) or \(x = -2\). In some questions you may be asked to give answers to a certain number of decimal places, but factorised roots are usually exact.

第三步：清晰写出解集：x = 5 或 x = −2。有些题目可能要求保留若干位小数，但因式分解得出的根通常是精确的。

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7. Special Cases: Difference of Two Squares | 特殊情况：平方差

A quadratic that has no middle term, such as \(x^2 – 9\), can be factored using the difference of two squares:

没有中间项的二次式，例如 x² − 9，可以使用平方差公式分解：

x² − a² = (x − a)(x + a)

Thus \(x^2 – 9 = (x – 3)(x + 3)\). Solving \(x^2 – 9 = 0\) gives \(x = 3\) or \(x = -3\). Always check if the constant term is a perfect square.

因此 x² − 9 = (x − 3)(x + 3)。解 x² − 9 = 0 得 x = 3 或 x = −3。始终检查常数项是否为完全平方数。

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8. Perfect Square Trinomials | 完全平方式

A perfect square trinomial has the form \(a^2 + 2ab + b^2\) or \(a^2 – 2ab + b^2\). For example, \(x^2 + 6x + 9 = (x + 3)^2\). Notice that \((x + 3)^2 = 0\) has only one distinct solution: \(x = -3\) (a repeated root).

完全平方式的形式为 a² + 2ab + b² 或 a² − 2ab + b²。例如，x² + 6x + 9 = (x + 3)²。注意 (x + 3)² = 0 只有唯一解：x = −3（重根）。

When you recognise a perfect square, the factorisation is quicker. Check if the first and last terms are perfect squares and whether the middle term is twice their product.

当你识别出完全平方时，因式分解会更快。检查首尾两项是否为完全平方，中间项是否为它们乘积的两倍。

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9. Checking Solutions by Substitution | 代入检验解

After solving, always check your answers by substituting them back into the original equation. For \(x = 5\) in \(x^2 – 3x – 10 = 0\): \(25 – 15 – 10 = 0\), correct. For \(x = -2\): \(4 + 6 – 10 = 0\), correct.

求解后，务必代入原方程检验。对于原方程 x² − 3x − 10 = 0，代入 x = 5：25 − 15 − 10 = 0，正确。代入 x = −2：4 + 6 − 10 = 0，正确。

Checking not only catches careless errors but also confirms that you have not missed a root. In a graphical context, the roots are the x-intercepts of the parabola.

检验不仅能发现粗心错误，还能确认没有漏根。在图像背景中，根是抛物线与 x 轴的交点。

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10. Common Mistakes and Tips | 常见错误与提示

• Forgetting to rearrange into standard form first. Always set one side to zero before factorising.

  忘记先将方程整理为标准形式。在因式分解前，务必使一边为零。

• Using the wrong signs when finding factors. Remember that if \(c\) is negative, the two factors have opposite signs.

  求因数时符号出错。记住，如果 c 是负数，两个因数符号相反。

• Dropping a solution when applying the zero product property. Write both linear equations separately.

  应用零乘积性质时漏掉一个解。请分别写出两个一次方程。

• Expanding incorrectly when checking. Expand carefully or use substitution to verify.

  检验时展开错误。请仔细展开或直接代入验证。

Tip: If a quadratic does not factorise easily with integers, you may need to use the quadratic formula or completing the square. However, many IGCSE factorisation questions are carefully selected to work with integers.

提示：如果二次式难以用整数分解，你可能需要使用求根公式或配方法。不过，许多 IGCSE 因式分解题都特意选择整数可分解的形式。

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11. Practice Questions | 练习题目

Try solving the following by factorisation:

尝试用因式分解法求解以下方程：

• \(x^2 + 7x + 12 = 0\)

  x² + 7x + 12 = 0

• \(x^2 – 4x – 21 = 0\)

  x² − 4x − 21 = 0

• \(2x^2 + 5x – 3 = 0\)

  2x² + 5x − 3 = 0

• \(9x^2 – 16 = 0\)

  9x² − 16 = 0

• \(x^2 – 10x + 25 = 0\)

  x² − 10x + 25 = 0

Answers: 1) –3, –4; 2) 7, –3; 3) 0.5, –3; 4) ±4/3; 5) 5 (repeated).

答案：1) −3, −4；2) 7, −3；3) 0.5, −3；4) ±4/3；5) 5（重根）。

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12. Summary: Key Steps for Solving by Factorisation | 总结：因式分解求解的关键步骤

To solve a quadratic equation by factorisation, follow these steps:

用因式分解法求解二次方程，遵循以下步骤：

• Rearrange into the form \(ax^2 + bx + c = 0\).

  整理成 ax² + bx + c = 0 的形式。

• Factorise the quadratic expression completely.

  将二次表达式完全因式分解。

• Set each factor equal to zero using the zero product property.

  利用零乘积性质，令每个因式等于零。

• Solve the resulting linear equations.

  解所得的一次方程。

• Check your answers by substitution.

  代入原方程检验答案。

Mastering factorisation is essential for success in IGCSE Mathematics, as it connects to graphs, algebraic fractions, and even calculus later. Practice consistently and you will build speed and accuracy.

掌握因式分解是取得 IGCSE 数学成功的关键，它与图像、代数分数乃至后面的微积分都有联系。坚持练习，你会变得既快又准。

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