Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与求根公式

📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与求根公式

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, coordinate geometry, and problem-solving questions. This article explains how to solve quadratic equations using three key methods: factorising, completing the square, and the quadratic formula. You will also learn how to use the discriminant and how to avoid common mistakes in the exam.

二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、坐标几何和问题解决类题目中。本文将讲解如何使用三种关键方法解二次方程：因式分解法、配方法和求根公式法。你还将学习如何使用判别式，以及如何在考试中避免常见错误。

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

A quadratic equation is a polynomial equation of degree 2. It contains at least one term that is squared, and the highest power of the variable is 2. The graph of a quadratic equation is a parabola, which is a smooth U-shaped or n-shaped curve.

二次方程是一个次数为 2 的多项式方程。它至少含有一个平方项，变量的最高次数为 2。二次方程的图像是一条抛物线，即一条光滑的 U 形或倒 U 形曲线。

In IGCSE mathematics, you will usually see quadratic equations written in terms of x, such as x² + 5x + 6 = 0. The solutions are the x-values where the parabola crosses the x-axis. These solutions are also called roots, zeros, or x-intercepts.

在 IGCSE 数学中，你通常会看到以 x 为变量的二次方程，例如 x² + 5x + 6 = 0。它的解就是抛物线与 x 轴交点的 x 坐标。这些解也被称为根、零点或 x 轴截距。

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

The standard form of a quadratic equation is written as:

二次方程的标准形式写作：

ax² + bx + c = 0

Here, a, b, and c are constants, and a must not be zero. If a = 0, the equation becomes linear rather than quadratic.

其中，a、b 和 c 是常数，且 a 不能为零。如果 a = 0，该方程就变成一次方程，而不是二次方程。

a is the coefficient of x² and cannot be zero.
a 是 x² 的系数，不能为零。
b is the coefficient of x.
b 是 x 的系数。
c is the constant term.
c 是常数项。

For example, in 3x² − 2x + 5 = 0, we have a = 3, b = −2, and c = 5. You should always rewrite an equation into standard form before choosing a solving method.

例如，在 3x² − 2x + 5 = 0 中，a = 3，b = −2，c = 5。在选择解题方法之前，你总是应该先把方程改写为标准形式。

3. Solving by Factorising | 因式分解法

Factorising is usually the quickest method when the quadratic expression can be written as a product of two linear brackets. If the product of two expressions equals zero, then at least one of the expressions must equal zero.

当二次表达式可以写成两个一次括号的乘积时，因式分解法通常是最快的方法。如果两个表达式的乘积等于零，那么其中至少有一个表达式必须等于零。

The key idea is the zero product property:

关键思想是零乘积性质：

If AB = 0, then A = 0 or B = 0

To solve x² + 5x + 6 = 0, first factorise the left-hand side into (x + 2)(x + 3). Then set each bracket equal to zero: x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3.

要解 x² + 5x + 6 = 0，先将左边因式分解为 (x + 2)(x + 3)。然后令每个括号分别等于零：x + 2 = 0 得 x = −2，x + 3 = 0 得 x = −3。

Always check your factors by expanding them. The factors must multiply back to the original quadratic expression. This helps you catch sign errors before writing your final answer.

总是要通过展开来检查你的因式。因式相乘后必须还原为原来的二次表达式。这样可以帮助你在写出最终答案之前发现符号错误。

4. Factorising When a ≠ 1 | 当 a ≠ 1 时的因式分解

When the coefficient of x² is not 1, factorising becomes slightly harder. For example, to factorise 2x² + 7x + 3, you need two numbers that multiply to give ac = 2 × 3 = 6 and add to give b = 7. Those numbers are 6 and 1.

当 x² 的系数不为 1 时，因式分解会稍微困难一些。例如，要对 2x² + 7x + 3 进行因式分解，你需要找到两个数，使它们的乘积为 ac = 2 × 3 = 6，和为 b = 7。这两个数是 6 和 1。

Split the middle term using these numbers:

使用这两个数拆分中间项：

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

Then set each bracket to zero. From (2x + 1) = 0, we get x = −1/2. From (x + 3) = 0, we get x = −3. This method is often called factorising by grouping or the ac method.

然后令每个括号分别等于零。由 (2x + 1) = 0，得 x = −1/2。由 (x + 3) = 0，得 x = −3。这种方法通常称为分组分解法或 ac 法。

If you cannot find two such integers easily, do not waste time guessing. Use the quadratic formula or completing the square instead.

如果你不能轻松找到这样的两个整数，不要浪费时间猜测。可以使用求根公式或配方法代替。

5. Completing the Square | 配方法

Completing the square is a powerful method that always works, even when the quadratic cannot be factorised easily. It is also useful for finding the vertex of a parabola and for deriving the quadratic formula.

配方法是一种始终有效的方法，即使二次方程不能轻松因式分解时也可以使用。它对于求抛物线的顶点和推导求根公式也非常有用。

To complete the square for x² + bx, add (b/2)² inside the bracket and subtract the same value outside to keep the expression balanced. The transformed form is:

要对 x² + bx 进行配方，在括号内加上 (b/2)²，并在括号外减去相同的值以保持表达式平衡。变换后的形式为：

x² + bx + c = (x + b/2)² − (b/2)² + c

For example, solve x² + 6x + 2 = 0 by completing the square. First write:

例如，用配方法解 x² + 6x + 2 = 0。首先写出：

x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7

Set this equal to zero: (x + 3)² − 7 = 0. Then (x + 3)² = 7, so x + 3 = ±√7. The solutions are x = −3 + √7 and x = −3 − √7.

令其等于零：(x + 3)² − 7 = 0。然后 (x + 3)² = 7，所以 x + 3 = ±√7。解为 x = −3 + √7 和 x = −3 − √7。

When a ≠ 1, first divide the whole equation by a, or factor a out of the first two terms before completing the square. This step is essential and is often where marks are lost.

当 a ≠ 1 时，先整式除以 a，或者先从变量项中提出 a，再进行配方。这个步骤至关重要，也是经常失分的地方。

6. The Quadratic Formula | 求根公式

The quadratic formula gives the solutions of any quadratic equation in standard form. It is especially useful when factorising is difficult or when the roots are irrational or involve fractions.

求根公式可以求出任何标准形式二次方程的解。当因式分解困难，或者根是无理数或含有分数时，它尤其有用。

For ax² + bx + c = 0, the solutions are:

对于 ax² + bx + c = 0，其解为：

x = (−b ± √(b² − 4ac)) / (2a)

Be careful with signs when substituting values. For example, in x² − 5x + 6 = 0, we have a = 1, b = −5, and c = 6. The formula gives x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2. The solutions are x = 3 and x = 2.

代入数值时要特别注意符号。例如，在 x² − 5x + 6 = 0 中，a = 1，b = −5，c = 6。公式给出 x = (5 ± √(25 − 24)) / 2 = (5 ± 1) / 2。解为 x = 3 和 x = 2。

Always write the formula before substituting. Clear substitution earns method marks even if a small arithmetic error occurs later.

一定要先写出公式再代入数值。清晰的代入过程即使后面出现小的计算错误，也能获得方法分。

7. Discriminant and Nature of Roots | 判别式与根的性质

The expression inside the square root of the quadratic formula is called the discriminant. It tells you how many real solutions a quadratic equation has without solving it fully.

求根公式中平方根内的表达式称为判别式。它可以在不解出完整方程的情况下，告诉你二次方程有多少个实数解。

D = b² − 4ac

If D > 0, the equation has two distinct real roots.
如果 D > 0，方程有两个不同的实数根。
If D = 0, the equation has one repeated real root, also called a double root.
如果 D = 0，方程有一个重复的实数根，也称为重根。
If D < 0, the equation has no real roots. In IGCSE, this means the graph does not cross the x-axis.
如果 D < 0，方程没有实数根。在 IGCSE 中，这意味着图像不与 x 轴相交。

For example, x² + 4x + 5 = 0 has D = 16 − 20 = −4, so it has no real solutions. You can state this without needing to find the roots.

例如，x² + 4x + 5 = 0 的判别式 D = 16 − 20 = −4，所以它没有实数解。你可以直接说明这一点，而不需要求出根。

Discriminant questions often appear as short-answer questions. Learn to calculate D quickly and interpret its meaning correctly.

判别式问题常常以简答题形式出现。要学会快速计算 D 并正确解释其意义。

8. Choosing the Best Method | 选择最佳方法

In the exam, you may be told which method to use, or you may need to choose. The best method depends on the form of the quadratic equation.

在考试中，题目可能会指定使用哪种方法，也可能需要你自己选择。最佳方法取决于二次方程的形式。

Use factorising if you can quickly find two factors that work.
如果能快速找到两个可行的因式，使用因式分解法。
Use completing the square when the question asks for the vertex form or exact surd answers.
当题目要求顶点式或精确根式答案时，使用配方法。
Use the quadratic formula when the coefficients are large, negative, or not easy to factorise.
当系数较大、为负数或不容易因式分解时，使用求根公式。

If the question says ‘give your answer correct to 2 decimal places’, the quadratic formula is often the best choice because it works directly with the coefficients.

如果题目要求 ‘答案精确到 2 位小数’，求根公式通常是最佳选择，因为它直接代入系数计算。

Whichever method you use, always substitute your solutions back into the original equation to check that they satisfy it.

无论使用哪种方法，都要将解代回原方程，检查它们是否满足原方程。

9. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many IGCSE students lose marks on quadratic equations because of small sign errors or careless arithmetic. Being aware of the most common mistakes can help you avoid them.

许多 IGCSE 学生在二次方程题上因符号错误或计算粗心而失分。了解最常见的错误可以帮助你避免它们。

Forgetting that a must not be zero in a quadratic equation.
忘记在二次方程中 a 不能为零。
Incorrectly expanding brackets when checking factorisation.
检查因式分解时错误地展开括号。
Misplacing the negative sign when substituting into the quadratic formula.
代入求根公式时放错负号位置。
Dividing by a when it is negative and losing the sign relationship.
当 a 为负数时除以 a，导致符号关系丢失。
Giving only one solution when the equation has two roots.
当方程有两个根时只给出一个解。

Always write down the standard form first. Then state your chosen method clearly. Show every step, especially the substitution into the quadratic formula or the completing the square process.

一定要先写出标准形式。然后清楚地说明你选择的方法。展示每一步，尤其是代入求根公式或配方法的过程。

Time management is also important. If factorising takes more than a few seconds, switch to the quadratic formula. It is reliable and avoids wasted minutes.

时间管理也很重要。如果因式分解超过几秒钟还没有思路，就改用求根公式。它可靠且不会浪费太多时间。

10. Worked Example | 例题精讲

Let us solve the equation 2x² − 3x − 5 = 0 using the quadratic formula. First identify a = 2, b = −3, and c = −5.

让我们使用求根公式解方程 2x² − 3x − 5 = 0。首先确定 a = 2，b = −3，c = −5。

Calculate the discriminant:

计算判别式：

D = (−3)² − 4(2)(−5) = 9 + 40 = 49

Since D = 49 is positive, there are two distinct real roots. Now substitute into the quadratic formula:

由于 D = 49 为正，有两个不同的实数根。现在代入求根公式：

x = (−(−3) ± √49) / (2 × 2) = (3 ± 7) / 4

This gives two solutions: x = (3 + 7) / 4 = 10 / 4 = 5 / 2, and x = (3 − 7) / 4 = −4 / 4 = −1.

得到两个解：x = (3 + 7) / 4 = 10 / 4 = 5 / 2，以及 x = (3 − 7) / 4 = −4 / 4 = −1。

Check the solutions by substitution. For x = 5/2, the expression 2(25/4) − 3(5/2) − 5 equals 25/2 − 15/2 − 10/2 = 0. For x = −1, the expression 2(1) + 3 − 5 = 0. Both roots are correct.

通过代入检查解。当 x = 5/2 时，表达式 2(25/4) − 3(5/2) − 5 等于 25/2 − 15/2 − 10/2 = 0。当 x = −1 时，表达式 2(1) + 3 − 5 = 0。两个根都正确。

This example shows the full process: standard form, discriminant, formula substitution, simplification, and checking. Repeating this structure will help you score full marks.

这个例子展示了完整过程：标准形式、判别式、公式代入、化简和检验。重复使用这个结构将帮助你获得满分。

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Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 解二次方程：因式分解、配方法与求根公式