Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most frequently tested topics in IGCSE Mathematics. You will encounter them in algebra, geometry, graphs, and real-world word problems. Mastering the three main methods — factorisation, completing the square, and the quadratic formula — is essential for exam success.

二次方程是 IGCSE 数学中考查最频繁的主题之一。你会在代数、几何、图像和实际应用题中遇到它们。掌握三种主要方法——因式分解法、配方法和二次公式——对于考试成功至关重要。

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

A quadratic equation is a polynomial equation of degree 2. Its general form is:

ax² + bx + c = 0, where a ≠ 0

Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The condition a ≠ 0 is crucial — if a = 0, the equation becomes linear.

二次方程是次数为 2 的多项式方程。它的一般形式为：

ax² + bx + c = 0，其中 a ≠ 0

其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 至关重要——如果 a = 0，方程就变成了线性方程。

Examples of quadratic equations:

x² − 5x + 6 = 0
2x² + 3x − 7 = 0
−x² + 4x = 0

二次方程的例子：

x² − 5x + 6 = 0
2x² + 3x − 7 = 0
−x² + 4x = 0

2. Roots and Solutions | 根与解

The values of x that make the equation true are called the roots (or solutions) of the quadratic equation. A quadratic equation can have two distinct real roots, one repeated real root, or no real roots — depending on the value of the discriminant.

使方程成立的 x 值称为二次方程的根（或解）。二次方程可以有两个不同的实数根、一个重根或没有实数根——这取决于判别式的值。

For example, in the equation x² − 3x + 2 = 0, substituting x = 1 gives 1 − 3 + 2 = 0, and substituting x = 2 gives 4 − 6 + 2 = 0. Therefore, the roots are x = 1 and x = 2.

例如，在方程 x² − 3x + 2 = 0 中，代入 x = 1 得 1 − 3 + 2 = 0，代入 x = 2 得 4 − 6 + 2 = 0。因此，根为 x = 1 和 x = 2。

3. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic can be easily factored. It relies on the zero product property: if the product of two expressions is zero, then at least one of them must be zero.

因式分解法是在二次式能够轻松分解时最快的方法。它依赖于零积性质：如果两个表达式的乘积为零，则其中至少有一个必须为零。

Step-by-step process:

Step 1: Write the equation in the form ax² + bx + c = 0.
Step 2: Factorise the quadratic expression.
Step 3: Set each bracket equal to zero.
Step 4: Solve each linear equation.

详细步骤：

第 1 步：将方程写成 ax² + bx + c = 0 的形式。
第 2 步：对二次表达式进行因式分解。
第 3 步：令每个括号等于零。
第 4 步：解每个线性方程。

Example: Solve x² − 5x + 6 = 0.

示例：解 x² − 5x + 6 = 0。

Find two numbers whose product is 6 and whose sum is −5. The numbers are −2 and −3 because (−2) × (−3) = 6 and (−2) + (−3) = −5.

寻找两个数，它们的乘积为 6，和为 −5。这两个数是 −2 和 −3，因为 (−2) × (−3) = 6 且 (−2) + (−3) = −5。

(x − 2)(x − 3) = 0

Using the zero product property:

x − 2 = 0 or x − 3 = 0

So the solutions are:

x = 2 or x = 3

根据零积性质：

x − 2 = 0 或 x − 3 = 0

因此解为：

x = 2 或 x = 3

4. Solving by Completing the Square | 配方法

Completing the square transforms a quadratic into a perfect square plus a constant. This method always works, even when factorisation is difficult.

配方法将一个二次式转化为一个完全平方加上一个常数。这种方法总是有效的，即使因式分解很困难时也能使用。

Example: Solve x² + 6x + 5 = 0.

示例：解 x² + 6x + 5 = 0。

Step 1: Move the constant term to the right-hand side.

x² + 6x = −5

第 1 步：将常数项移到右边。

x² + 6x = −5

Step 2: Take half of the coefficient of x (which is 6), giving 3, and square it to get 9. Add 9 to both sides.

x² + 6x + 9 = −5 + 9

第 2 步：取 x 系数（6）的一半得 3，再平方得 9。两边同时加 9。

x² + 6x + 9 = −5 + 9

Step 3: The left-hand side is now a perfect square.

(x + 3)² = 4

第 3 步：左边现在是一个完全平方。

(x + 3)² = 4

Step 4: Take the square root of both sides. Remember to include the ± sign.

x + 3 = ±2

第 4 步：两边开平方。记住要加上 ± 号。

x + 3 = ±2

Thus x = −3 + 2 = −1 or x = −3 − 2 = −5.

因此 x = −3 + 2 = −1 或 x = −3 − 2 = −5。

5. The Quadratic Formula | 二次公式

The quadratic formula is a universal method that solves any quadratic equation. It is derived from completing the square and is especially useful when factorisation is not straightforward.

二次公式是一种通用的方法，可以解任何二次方程。它由配方法推导而来，特别适用于因式分解不直接的情况。

x = [−b ± √(b² − 4ac)] / (2a)

To use the formula, first identify a, b, and c from the equation ax² + bx + c = 0, then substitute them into the formula.

要使用该公式，首先从方程 ax² + bx + c = 0 中确定 a、b 和 c，然后将它们代入公式。

Example: Solve 2x² − 7x + 3 = 0 using the quadratic formula.

示例：使用二次公式解 2x² − 7x + 3 = 0。

Here, a = 2, b = −7, c = 3. Substituting into the formula:

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

这里，a = 2，b = −7，c = 3。代入公式得：

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

x = [7 ± √(49 − 24)] / 4 = [7 ± √25] / 4 = (7 ± 5) / 4

Thus:

x = (7 + 5) / 4 = 12 / 4 = 3 or x = (7 − 5) / 4 = 2 / 4 = ½

因此：

x = (7 + 5) / 4 = 12 / 4 = 3 或 x = (7 − 5) / 4 = 2 / 4 = ½

6. The Discriminant | 判别式

The expression b² − 4ac inside the square root is called the discriminant, denoted by Δ. It tells us how many real roots the equation has without solving it fully.

根号内的表达式 b² − 4ac 称为判别式，用 Δ 表示。它告诉我们方程有多少个实数根，而无需完整求解。

Discriminant 判别式	Number of Real Roots 实数根的个数
Δ > 0	Two distinct real roots 两个不同的实数根
Δ = 0	One repeated real root 一个重根
Δ < 0	No real roots 没有实数根

Example: Determine the number of roots of x² − 4x + 4 = 0.

示例：确定 x² − 4x + 4 = 0 的根的个数。

Here, a = 1, b = −4, c = 4. So Δ = (−4)² − 4 × 1 × 4 = 16 − 16 = 0. Since Δ = 0, the equation has exactly one repeated real root.

这里，a = 1，b = −4，c = 4。因此 Δ = (−4)² − 4 × 1 × 4 = 16 − 16 = 0。因为 Δ = 0，该方程恰好有一个重根。

7. Solving Word Problems | 解应用题

Quadratic equations often appear in geometry and number problems. The key steps are: define a variable, write an equation, solve it, and check whether the answer makes sense in context.

二次方程经常出现在几何和数字问题中。关键步骤是：设一个变量、写出方程、求解，并检查答案在上下文中是否合理。

Example: A rectangle has an area of 40 cm². Its length is 3 cm longer than its width. Find the width.

示例：一个矩形的面积为 40 cm²。它的长度比宽度长 3 cm。求宽度。

Let the width be w cm. Then the length is (w + 3) cm. The area equation is:

w(w + 3) = 40

设宽度为 w cm。则长度为 (w + 3) cm。面积方程为：

w(w + 3) = 40

Expanding and rearranging:

w² + 3w − 40 = 0

展开并整理得：

w² + 3w − 40 = 0

Factorising: (w + 8)(w − 5) = 0, so w = −8 or w = 5. Since width cannot be negative, w = 5 cm.

因式分解：(w + 8)(w − 5) = 0，所以 w = −8 或 w = 5。由于宽度不能为负，w = 5 cm。

8. Common Mistakes | 常见错误

Many students lose marks on quadratic equations due to avoidable errors. Here are the most common pitfalls.

许多学生在二次方程上丢分是因为一些可以避免的错误。以下是最常见的陷阱。

Forgetting to rearrange: Always write the equation in the form ax² + bx + c = 0 before factorising or using the formula.
Missing the ± sign: When taking a square root, remember that both positive and negative roots are possible.
Incorrect signs: Be careful with negative values of a, b, and c when substituting into the formula.
Rejecting valid negative roots: In pure algebra, negative roots are valid; only reject them if the context demands a positive quantity.

忘记整理方程：在因式分解或使用公式之前，一定要把方程写成 ax² + bx + c = 0 的形式。

漏掉 ± 号：取平方根时，要记住正根和负根都有可能。

符号错误：将 a、b、c 代入公式时要特别小心负值。

错误地舍弃负根：在纯代数中，负根是有效的；只有在题目背景要求正量时才舍弃负根。

9. Practice Questions | 练习题目

Try these questions on your own before checking the answers below.

请先自己尝试以下题目，再核对下方的答案。

Question 题目	Answer 答案
1. Solve x² − 3x − 10 = 0	x = 5 or x = −2
2. Solve 2x² − 5x − 7 = 0	x = 3.5 or x = −1
3. Solve x² + 8x + 16 = 0	x = −4 (repeated)
4. Find the discriminant of 3x² − 6x + 2 = 0	Δ = 12, two real roots

Worked answer for Question 2: 2x² − 5x − 7 = 0. Factorising gives (2x − 7)(x + 1) = 0, so 2x − 7 = 0 or x + 1 = 0. Therefore x = 7/2 = 3.5 or x = −1.

第 2 题详细解答：2x² − 5x − 7 = 0。因式分解得 (2x − 7)(x + 1) = 0，所以 2x − 7 = 0 或 x + 1 = 0。因此 x = 7/2 = 3.5 或 x = −1。

10. Summary | 总结

To solve a quadratic equation successfully, follow these steps:

要成功解二次方程，请遵循以下步骤：

Rearrange the equation into the standard form ax² + bx + c = 0.
Choose the most efficient method: factorise if possible, otherwise use completing the square or the quadratic formula.
Use the discriminant b² − 4ac to check the number of real roots.
Always check your solutions by substituting them back into the original equation.

将方程整理为标准形式 ax² + bx + c = 0。

选择最高效的方法：如果可以，优先因式分解；否则使用配方法或二次公式。

使用判别式 b² − 4ac 来检查实数根的个数。

始终将解代回原方程进行验证。

With regular practice, solving quadratic equations becomes quick and accurate. This skill will serve you well across all areas of the IGCSE Mathematics syllabus.

通过定期练习，解二次方程会变得快速而准确。这项技能将使你在 IGCSE 数学课程的所有领域中受益。

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Solving Quadratic Equations | 解二次方程