📚 Mastering Quadratic Equations for IGCSE Mathematics | 掌握IGCSE数学二次方程
Quadratic equations are a central topic in IGCSE Mathematics, appearing in algebra, graphs, and problem-solving questions. This article explains the standard form of a quadratic equation and demonstrates three reliable solution methods: factorisation, completing the square, and the quadratic formula. You will also learn how the discriminant reveals the nature of the roots and how to avoid common exam mistakes.
二次方程是IGCSE数学中的一个核心主题，广泛出现在代数、图像和应用题中。本文讲解二次方程的标准形式，并演示三种可靠的求解方法：因式分解法、配方法和二次公式法。你还将学习判别式如何揭示根的性质，以及如何避免常见考试错误。
1. What Is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is any equation that can be written in the general form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the unknown variable x is 2, which gives the equation its name.
二次方程是任何可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a ≠ 0。未知变量 x 的最高次数是2，这正是该方程名称的由来。
For example, 3x² − 7x + 2 = 0 is quadratic because the highest exponent is 2. Equations such as 5x − 8 = 0 are linear, not quadratic, because the variable only appears to the first power.
例如，3x² − 7x + 2 = 0 是二次方程，因为最高指数是2。像 5x − 8 = 0 这样的方程是线性方程，而不是二次方程，因为变量只出现一次方。
2. Standard Form and Key Terms | 标准形式与关键术语
Before solving a quadratic equation, you should rearrange it into standard form. This means all terms are on one side of the equation, equal to zero, and are usually ordered by descending powers of x.
在求解二次方程之前，你应将其整理为标准形式。这意味着所有项都移到方程的一边，等于零，并且通常按 x 的降幂排列。
ax² + bx + c = 0
Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The values x that make the equation true are called the roots or solutions of the equation.
这里，a 是 x² 的系数，b 是 x 的系数，c 是常数项。使方程成立的 x 值称为方程的根或解。
For instance, 2x² − 3x + 1 = 0 has a = 2, b = −3, and c = 1. A quadratic equation normally has two solutions, although they may be equal or may not be real numbers.
例如，2x² − 3x + 1 = 0 中 a = 2、b = −3、c = 1。二次方程通常有两个解，尽管它们可能相等，也可能不是实数。
3. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when the quadratic expression factorises neatly into two linear brackets. The method relies on the fact that if two factors multiply to give zero, then at least one of the factors must equal zero.
当二次表达式可以整齐地分解为两个一次因式时，因式分解法通常是最快的方法。该方法依据的原理是：如果两个因式的乘积为零，那么至少有一个因式必须等于零。
To solve x² + bx + c = 0, find two numbers that multiply to give c and add to give b. These numbers become the constants in the brackets.
要求解 x² + bx + c = 0，需要找出两个数，它们的乘积等于 c，和等于 b。这两个数就成为括号中的常数项。
x² + bx + c = (x + p)(x + q)
Once factorised, set each bracket equal to zero and solve for x. This gives the two roots directly.
分解完成后，令每个括号分别等于零并解出 x。这样可以直接得到两个根。
4. Worked Example: Factorisation | 实例：因式分解
Solve the equation x² − 5x + 6 = 0 by factorisation.
用因式分解法解方程 x² − 5x + 6 = 0。
We need two numbers that multiply to +6 and add to −5. The numbers are −2 and −3, so the factorised form is (x − 2)(x − 3) = 0.
我们需要两个数，乘积为 +6，和为 −5。这两个数是 −2 和 −3，因此因式分解形式为 (x − 2)(x − 3) = 0。
(x − 2)(x − 3) = 0
Setting each factor to zero gives x − 2 = 0 or x − 3 = 0, so the solutions are x = 2 and x = 3.
令每个因式等于零，得到 x − 2 = 0 或 x − 3 = 0，因此解为 x = 2 和 x = 3。
Always check your solutions by substituting them back into the original equation. For x = 2, the left side is 4 − 10 + 6 = 0, and for x = 3, it is 9 − 15 + 6 = 0.
一定要将解代回原方程进行检验。对于 x = 2，左边为 4 − 10 + 6 = 0；对于 x = 3，左边为 9 − 15 + 6 = 0。
5. Completing the Square | 配方法
Completing the square is a powerful method that works for all quadratic equations, even when factorisation is difficult. The goal is to rewrite the expression ax² + bx + c in the form a(x + p)² + q, from which the roots can be found by taking square roots.
配方法是一种强大的方法，适用于所有二次方程，即使因式分解很困难时也能使用。其目标是将表达式 ax² + bx + c 改写为 a(x + p)² + q 的形式，然后通过开平方求出根。
For x² + bx, the completed square form is written as (x + b/2)² − (b/2)². This is based on the identity (x + p)² = x² + 2px + p².
对于 x² + bx，配方后的形式为 (x + b/2)² − (b/2)²。这是基于恒等式 (x + p)² = x² + 2px + p²。
x² + bx = (x + b/2)² − (b/2)²
After rewriting, set the expression equal to zero, isolate the squared bracket, and take the positive and negative square roots.
改写后，令表达式等于零，分离平方括号，然后取正负平方根。
6. Worked Example: Completing the Square | 实例：配方法
Solve x² + 6x − 7 = 0 by completing the square.
用配方法解方程 x² + 6x − 7 = 0。
Start with the x² + 6x part. Half of the coefficient of x is 3, so we write x² + 6x = (x + 3)² − 9. The original equation becomes (x + 3)² − 9 − 7 = 0.
先处理 x² + 6x 部分。x 系数的一半是 3，所以写成 x² + 6x = (x + 3)² − 9。原方程变为 (x + 3)² − 9 − 7 = 0。
(x + 3)² − 16 = 0
Add 16 to both sides: (x + 3)² = 16. Taking the square root of both sides gives x + 3 = ±4.
两边同时加16：(x + 3)² = 16。对两边开平方得到 x + 3 = ±4。
Therefore, x = −3 + 4 = 1 or x = −3 − 4 = −7. The solutions are x = 1 and x = −7.
因此，x = −3 + 4 = 1 或 x = −3 − 4 = −7。解为 x = 1 和 x = −7。
7. The Quadratic Formula | 二次公式
The quadratic formula gives the solutions of any quadratic equation ax² + bx + c = 0 directly. It is especially useful when the coefficients are not easy to factorise.
二次公式可以直接给出任何二次方程 ax² + bx + c = 0 的解。当系数不容易因式分解时，它尤其有用。
x = (−b ± √b² − 4ac) ÷ 2a
The symbol ± means that there are generally two solutions: one using the plus sign and one using the minus sign. The expression under the square root, b² − 4ac, is called the discriminant.
符号 ± 表示通常有两个解：一个使用加号，一个使用减号。平方根下的表达式 b² − 4ac 称为判别式。
Before using the formula, identify a, b and c carefully, especially when there are negative signs.
使用公式前，要仔细确定 a、b 和 c，尤其是存在负号时。
8. Worked Example: Quadratic Formula | 实例：二次公式
Solve 2x² + 3x − 2 = 0 using the quadratic formula.
使用二次公式解方程 2x² + 3x − 2 = 0。
Here, a = 2, b = 3, and c = −2. Substitute these into the formula:
这里，a = 2、b = 3、c = −2。将它们代入公式：
x = (−3 ± √(3² − 4 × 2 × (−2))) ÷ (2 × 2)
Simplify inside the square root: 3² − 4 × 2 × (−2) = 9 + 16 = 25. So the formula becomes x = (−3 ± √25) ÷ 4.
化简平方根内部：3² − 4 × 2 × (−2) = 9 + 16 = 25。因此公式变为 x = (−3 ± √25) ÷ 4。
x = (−3 ± 5) ÷ 4
Using the plus sign gives x = (2) ÷ 4 = 1/2. Using the minus sign gives x = (−8) ÷ 4 = −2. The solutions are x = 1/2 and x = −2.
使用加号得到 x = (2) ÷ 4 = 1/2。使用减号得到 x = (−8) ÷ 4 = −2。解为 x = 1/2 和 x = −2。
9. Discriminant and Nature of Roots | 判别式与根的性质
The discriminant Δ = b² − 4ac tells you how many real solutions a quadratic equation has, without needing to solve it completely.
判别式 Δ = b² − 4ac 可以告诉你二次方程有多少个实数解，而无需完全求解。
| Discriminant value | Nature of roots |
|---|---|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One repeated real root |
| Δ < 0 | No real roots |
This is particularly useful in exam questions that ask you to state the number of solutions or to determine whether a graph meets the x-axis.
这在考试题中特别有用，例如要求你说明解的个数，或判断图像是否与x轴相交。
For x² − 5x + 6 = 0, the discriminant is 25 − 24 = 1, which is positive, so there are two distinct real roots: x = 2 and x = 3.
对于 x² − 5x + 6 = 0，判别式为 25 − 24 = 1，是正数，因此有两个不同的实数根：x = 2 和 x = 3。
10. Graphical Interpretation | 图像解释
The graph of y = ax² + bx + c is a parabola. The solutions of ax² + bx + c = 0 correspond to the x-coordinates where the parabola crosses the x-axis.
y = ax² + bx + c 的图像是一条抛物线。方程 ax² + bx + c = 0 的解对应于抛物线与x轴交点的横坐标。
If the discriminant is positive, the parabola crosses the x-axis at two points. If it is zero, the parabola just touches the x-axis at one point. If it is negative, the parabola does not meet the x-axis at all.
如果判别式为正，抛物线与x轴相交于两点。如果判别式为零，抛物线仅在一点与x轴相切。如果判别式为负，抛物线与x轴完全不相交。
For example, y = x² − 4x + 3 factorises to y = (x − 1)(x − 3), so the graph crosses the x-axis at x = 1 and x = 3.
例如，y = x² − 4x + 3 可以因式分解为 y = (x − 1)(x − 3)，因此图像在 x = 1 和 x = 3 处穿过x轴。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is forgetting to rearrange the equation into standard form before identifying a, b and c. Another is making sign errors when substituting negative coefficients into the quadratic formula.
一个常见错误是在确定 a、b 和 c 之前忘记将方程整理为标准形式。另一个常见错误是在将负系数代入二次公式时出现符号错误。
- Always write ax² + bx + c = 0 first, with zero on one side.
- Show every step in your working, especially when simplifying the square root part of the formula.
- Check your answers by substituting them back into the original equation.
- If factorisation seems difficult, use the quadratic formula or completing the square instead.
- 一定要先写成 ax² + bx + c = 0 的形式，使一边为零。
- 每一步都要写出过程，尤其是在化简公式中平方根部分时。
- 将答案代回原方程进行检验。
- 如果因式分解看起来困难，可以改用二次公式或配方法。
In IGCSE exams, marks are often awarded for correct substitution and working, even if the final answer is slightly wrong, so always write down your method clearly.
在IGCSE考试中，即使最终答案略有错误，正确的代入和解题过程也常常能得分，因此一定要清晰地写出你的方法。
12. Summary of Methods | 方法总结
To solve a quadratic equation, first bring it to standard form. Then choose the most efficient method: factorisation for simple quadratics, completing the square for exact forms or proof, and the quadratic formula for any equation.
解二次方程时，首先要将其化为标准形式。然后选择最有效的方法：简单的二次方程用因式分解法，需要精确形式或证明时用配方法，任何方程都可以用二次公式法。
Factorisation → (x + p)(x + q) = 0
Completing the square → a(x + p)² + q = 0
Quadratic formula → x = (−b ± √b² − 4ac) ÷ 2a
Practise all three methods so that you can switch between them confidently in the exam. Regular practice with past paper questions will improve both speed and accuracy.
练习所有三种方法，以便在考试中能够自信地在它们之间切换。经常练习历年真题将提高速度和准确性。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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