Solving Quadratic Equations: Factorising, Formula, and Completing the Square | 二次方程求解：因式分解、公式法与配方法

📚 Solving Quadratic Equations: Factorising, Formula, and Completing the Square | 二次方程求解：因式分解、公式法与配方法

Quadratic equations appear frequently in the Edexcel IGCSE Mathematics exams. In this revision article, we explore the standard form of a quadratic equation and the three core algebraic methods for solving it: factorising, the quadratic formula, and completing the square. We also examine the discriminant, use graphs to interpret roots, and apply all these ideas to practical word problems.

二次方程在 Edexcel IGCSE 数学考试中频繁出现。在这篇复习文章中，我们探讨二次方程的标准形式以及求解它的三种核心代数方法：因式分解、求根公式和配方法。我们还会研究判别式，利用图像解释根的意义，并将这些思想应用于实际应用题。

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

A quadratic equation is an equation in which the highest power of the variable is 2. Its general form is

二次方程是变量的最高次数为 2 的方程。它的一般形式为

ax² + bx + c = 0, where a ≠ 0

Here a, b and c are constants. The coefficient a cannot be zero, because then the equation would become linear. Examples of quadratic equations include 2x² − 3x + 1 = 0 and x² + 4x − 5 = 0.

其中 a、b、c 是常数。系数 a 不能为零，因为那样方程就会变成一次方程。二次方程的例子包括 2x² − 3x + 1 = 0 和 x² + 4x − 5 = 0。

A solution of a quadratic equation is called a root. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots, depending on the value of the discriminant. We will discuss the discriminant in Section 6.

二次方程的解称为根。二次方程可以有两个不同的实数根、一个重根，或者没有实数根，这取决于判别式的取值。我们将在第 6 节中讨论判别式。

2. Factorising Quadratic Expressions | 二次多项式的因式分解

Before solving a quadratic equation by factorising, we must be able to factorise a quadratic expression. For a quadratic of the form x² + bx + c, we look for two numbers whose product is c and whose sum is b.

在用因式分解法求解二次方程之前，我们必须能够分解二次多项式。对于形如 x² + bx + c 的二次式，我们要找两个数，使得它们的乘积等于 c，并且它们的和等于 b。

For example, to factorise x² + 5x + 6, we need two numbers that multiply to 6 and add to 5. These numbers are 2 and 3, so we write

例如，要分解 x² + 5x + 6，我们需要找两个数，它们相乘等于 6，相加等于 5。这两个数是 2 和 3，因此我们可以写成

x² + 5x + 6 = (x + 2)(x + 3)

When the coefficient of x² is not 1, for example 2x² + 7x + 3, we can use the method of splitting the middle term. Multiply a and c: 2 × 3 = 6. Then find two numbers that multiply to 6 and add to 7. The numbers are 1 and 6, so we split 7x into 1x + 6x.

当 x² 的系数不是 1 时，例如 2x² + 7x + 3，我们可以使用裂项法。将 a 与 c 相乘：2 × 3 = 6。然后找两个数，它们相乘等于 6，相加等于 7。这两个数是 1 和 6，因此把 7x 拆成 1x + 6x。

2x² + 7x + 3 = 2x² + 1x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1)

Check your factorisation by expanding the brackets. This is also an excellent way to avoid sign errors.

可以通过展开括号来检查分解是否正确。这也是避免符号错误的一个好方法。

3. Solving by Factorising | 用因式分解法解方程

To solve a quadratic equation by factorising, first rearrange it so that one side is zero. Then factorise the quadratic expression and use the zero-product property: if A × B = 0, then A = 0 or B = 0.

用因式分解法解二次方程时，先整理方程，使一边为零。然后分解二次多项式，并利用零乘积性质：如果 A × B = 0，那么 A = 0 或 B = 0。

Consider the equation x² − 3x − 10 = 0. The two numbers that multiply to −10 and add to −3 are −5 and 2. Therefore

考虑方程 x² − 3x − 10 = 0。两个相乘等于 −10 且相加等于 −3 的数是 −5 和 2。因此

x² − 3x − 10 = (x − 5)(x + 2) = 0

Setting each factor to zero gives x − 5 = 0 or x + 2 = 0, so the roots are x = 5 and x = −2.

令每个因式等于零，得到 x − 5 = 0 或 x + 2 = 0，所以根为 x = 5 和 x = −2。

Always check whether the equation needs rearranging first. For example, 2x² = 5x + 3 must be rewritten as 2x² − 5x − 3 = 0 before factorising.

始终注意方程是否需要先移项。例如，2x² = 5x + 3 必须先改写为 2x² − 5x − 3 = 0，然后再分解。

4. The Quadratic Formula | 求根公式

Not every quadratic expression factorises easily, and some quadratic equations cannot be solved by simple integer factorisation. In those cases, we use the quadratic formula, which gives the roots of any quadratic equation ax² + bx + c = 0.

并非所有二次多项式都能轻松因式分解，有些二次方程无法通过简单的整数因式分解求解。在这种情况下，我们使用求根公式，它给出任何二次方程 ax² + bx + c = 0 的根。

x = (−b ± √(b² − 4ac)) / (2a)

For example, solve 2x² + 3x − 2 = 0. Here a = 2, b = 3, c = −2. Substitute these values into the formula.

例如，求解 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。将这些值代入公式。

x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √(9 + 16)) / 4 = (−3 ± √25) / 4 = (−3 ± 5) / 4

This gives x = (−3 + 5)/4 = 1/2 or x = (−3 − 5)/4 = −2. So the solutions are x = 1/2 and x = −2.

这得到 x = (−3 + 5)/4 = 1/2 或 x = (−3 − 5)/4 = −2。因此解为 x = 1/2 和 x = −2。

You must be confident substituting negative numbers into the formula, especially when b or c is negative. Many marks are lost because of sign errors.

你必须能够熟练地将负数代入公式，尤其当 b 或 c 为负数时。许多分数都因符号错误而丢失。

5. Solving by Completing the Square | 配方法

Completing the square is another algebraic method that rewrites a quadratic expression in the form p(x + q)² + r. It is particularly useful when the quadratic cannot be factorised, and it also helps identify the vertex of a parabola.

配方法是另一种代数方法，它将二次多项式改写为 p(x + q)² + r 的形式。当二次式无法因式分解时，这种方法尤其有用，它也有助于确定抛物线的顶点。

Start with the equation x² + 6x + 4 = 0. Take half of the coefficient of x, which is 3, and square it: 9. Then rewrite the expression as

从方程 x² + 6x + 4 = 0 开始。取 x 系数的半为 3，并平方得 9。然后将表达式改写为

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

So the equation becomes (x + 3)² − 5 = 0, or (x + 3)² = 5. Taking square roots gives x + 3 = ±√5, so

于是方程变为 (x + 3)² − 5 = 0，即 (x + 3)² = 5。两边开平方得 x + 3 = ±√5，所以

x = −3 ± √5

If the coefficient of x² is not 1, divide the whole equation by that coefficient first. For example, 2x² + 8x + 3 = 0 is first written as x² + 4x + 3/2 = 0, then completed as before.

如果 x² 的系数不是 1，先将整个方程除以该系数。例如，2x² + 8x + 3 = 0 首先写成 x² + 4x + 3/2 = 0，然后再按上面的方法配方。

6. The Discriminant | 判别式

The expression inside the square root of the quadratic formula is called the discriminant. It is denoted by Δ, where Δ = b² − 4ac.

求根公式中根号内的表达式称为判别式，记为 Δ，其中 Δ = b² − 4ac。

The value of the discriminant tells us how many real roots a quadratic equation has.

判别式的值告诉我们二次方程有多少个实数根。

If Δ > 0, the equation has two distinct real roots.

如果 Δ > 0，方程有两个不同的实数根。
If Δ = 0, the equation has one repeated real root.

如果 Δ = 0，方程有一个实数重根。
If Δ < 0, the equation has no real roots.

如果 Δ < 0，方程没有实数根。

For example, in x² + 4x + 5 = 0, we have Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4, so there are no real roots.

例如，在 x² + 4x + 5 = 0 中，有 Δ = 4² − 4 × 1 × 5 = 16 − 20 = −4，因此没有实数根。

In Edexcel IGCSE questions, you may be asked to show that an equation has no real roots or to find the range of values of k for which an equation has two real roots.

在 Edexcel IGCSE 考试中，你可能会被要求证明方程没有实数根，或者求某个参数 k 的取值范围使方程有两个实数根。

7. Graphs of Quadratic Functions | 二次函数图像

The graph of a quadratic function y = ax² + bx + c is a parabola. If a > 0, the parabola opens upwards (U shape); if a < 0, it opens downwards (n shape).

二次函数 y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上（U 形）；如果 a < 0，抛物线开口向下（n 形）。

The x-coordinate of the vertex is given by x = −b/(2a). To find the y-coordinate, substitute this x-value back into the equation.

顶点的 x 坐标为 x = −b/(2a)。要求 y 坐标，只需将该 x 值代入原方程。

When drawing a quadratic graph, the roots correspond to the x-intercepts. The discriminant tells you whether the graph crosses the x-axis twice, touches it once, or never meets it.

绘制二次函数图像时，根对应 x 轴截距。判别式告诉你图像是与 x 轴相交两次、相切一次，还是完全不相交。

y = (x − 1)² − 4 has vertex (1, −4) and roots x = −1 and x = 3

Completing the square is a direct way to read the vertex: in y = (x − p)² + q, the vertex is (p, q).

配方法是直接读取顶点的方法：在 y = (x − p)² + q 中，顶点为 (p, q)。

8. Word Problems with Quadratics | 二次方程应用题

Quadratic equations often appear in real-life problems such as areas, profits, and projectile motion. The key is to translate the words into an equation with a single variable, then solve it using one of the methods above.

二次方程经常出现在实际应用中，例如面积、利润和抛体运动。关键在于将文字翻译成含一个变量的方程，然后用上面的某种方法求解。

Example: A rectangle has length 4 cm longer than its width. Its area is 96 cm². Find the width.

例如：一个矩形的长比宽长 4 cm。它的面积是 96 cm²。求宽。

Let the width be w cm. Then the length is (w + 4) cm. The area equation is

设宽为 w cm。则长为 (w + 4) cm。面积方程为

w(w + 4) = 96 ⟹ w² + 4w − 96 = 0

Factorise: (w + 12)(w − 8) = 0. Thus w = 8 or w = −12. Since a width cannot be negative, the width is 8 cm.

分解因式：(w + 12)(w − 8) = 0。因此 w = 8 或 w = −12。由于宽不能为负数，所以宽为 8 cm。

In word problems, always reject negative solutions if they do not make sense in the context, and clearly state the units in your final answer.

在应用题中，如果负解不符合实际情况，一定要舍去，并在最终答案中明确写出单位。

9. Common Mistakes | 常见错误

Students often make the same errors when solving quadratic equations. Avoiding these pitfalls will help you earn full marks.

学生在求解二次方程时经常犯相似的错误。避免这些陷阱将帮助你获得满分。

Forgetting to rearrange the equation into the form ax² + bx + c = 0 before factorising or using the formula.

在因式分解或使用公式前，忘记将方程整理成 ax² + bx + c = 0 的形式。
Losing negative signs when substituting into the quadratic formula. Write each step clearly and use brackets around negative values.

代入求根公式时丢失负号。要把每一步写清楚，并在负数周围加上括号。
Dividing both sides by x when x is a possible root. This can lose a solution, such as x = 0.

方程两边同时除以 x，这可能丢掉解，例如 x = 0。
Confusing (x + 3)² with x² + 9. Remember that (x + 3)² = x² + 6x + 9.

把 (x + 3)² 与 x² + 9 混淆。记住 (x + 3)² = x² + 6x + 9。
Incorrectly stating “no solutions” when the discriminant is negative. For IGCSE, it means no real solutions.

当判别式为负时错误地说“无解”。在 IGCSE 中，这意味着没有实数解。

10. Practice Questions | 练习题

Try these questions on your own before checking the answers. They cover the main skills from this article.

先独立尝试这些题目，然后再核对答案。它们涵盖了本文的主要技巧。

Question 1: Solve x² + 7x + 12 = 0 by factorising.

问题 1：用因式分解法解 x² + 7x + 12 = 0。

Question 2: Use the quadratic formula to solve 3x² − 5x − 2 = 0.

问题 2：用求根公式解 3x² − 5x − 2 = 0。

Question 3: Solve by completing the square: x² + 8x + 10 = 0.

问题 3：用配方法解 x² + 8x + 10 = 0。

Question 4: Find the value of the discriminant for 2x² + 6x + 5, and state how many real roots the equation has.

问题 4：求 2x² + 6x + 5 的判别式的值，并说明该方程有多少个实数根。

Answers: 1. x = −3 or x = −4. 2. x = −1/3 or x = 2. 3. x = −4 ± √6. 4. Δ = 36 − 40 = −4, so no real roots.

答案：1. x = −3 或 x = −4。 2. x = −1/3 或 x = 2。 3. x = −4 ± √6。 4. Δ = 36 − 40 = −4，所以没有实数根。

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Solving Quadratic Equations: Factorising, Formula, and Completing the Square | 二次方程求解：因式分解、公式法与配方法