📚 Solving Quadratic Equations | 一元二次方程的解法
Quadratic equations are a central topic in the Edexcel IGCSE Mathematics course. You will meet them in both Foundation and Higher tier papers, and they appear in many real-world problems. Understanding how to solve them — and knowing which method to choose — is essential for exam success.
一元二次方程是 Edexcel IGCSE 数学课程的核心内容。无论是基础级还是高级试卷都会出现，并且许多实际问题都会用到它们。理解方程的各种解法，并知道在什么情况下选择哪种方法，是考试成功的关键。
1. What Is a Quadratic Equation? | 什么是一元二次方程？
A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the unknown x is 2, which gives the equation its name. Examples include x² – 5x + 6 = 0 and 2x² + 3x – 1 = 0.
一元二次方程是形如 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。未知数 x 的最高次数是 2，这也是“二次”名称的由来。例如 x² – 5x + 6 = 0 和 2x² + 3x – 1 = 0。
2. Expanding and Factorising | 展开与因式分解
Before solving, you must be comfortable with expanding brackets and factorising. To expand, multiply every term in the first bracket by every term in the second bracket. For example: (x + 3)(x – 2) = x² + x – 6.
在求解之前，你需要熟练展开括号和因式分解。展开时，将第一个括号中的每一项与第二个括号中的每一项相乘。例如：(x + 3)(x – 2) = x² + x – 6。
Factorising is the reverse process. To factorise a quadratic expression like x² + bx + c, find two numbers that multiply to give c and add to give b. For x² – 4x + 3, the numbers -1 and -3 work because (-1) × (-3) = 3 and (-1) + (-3) = -4. So x² – 4x + 3 = (x – 1)(x – 3).
因式分解是相反的过程。对于形如 x² + bx + c 的二次表达式，找出两个数，使它们的乘积为 c，和为 b。对于 x² – 4x + 3，-1 和 -3 满足条件，因为 (-1) × (-3) = 3，(-1) + (-3) = -4。所以 x² – 4x + 3 = (x – 1)(x – 3)。
3. Solving by Factorising | 用因式分解法求解
If a quadratic equation can be factorised, the zero product property says that if ab = 0, then a = 0 or b = 0. For example, solve x² + 5x + 6 = 0. Factorise to get (x + 2)(x + 3) = 0. Then either x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3.
如果一个一元二次方程可以因式分解，那么根据零因子性质：若 ab = 0，则 a = 0 或 b = 0。例如，解 x² + 5x + 6 = 0。因式分解得 (x + 2)(x + 3) = 0。于是 x + 2 = 0 或 x + 3 = 0，所以 x = -2 或 x = -3。
For equations with a leading coefficient greater than 1, such as 2x² – 7x + 3 = 0, you need to factorise carefully. The factorisation is (2x – 1)(x – 3) = 0. This gives x = 1/2 or x = 3.
对于首项系数大于 1 的方程，例如 2x² – 7x + 3 = 0，需要仔细因式分解。其分解为 (2x – 1)(x – 3) = 0，从而得到 x = 1/2 或 x = 3。
For ax² + bx + c = 0, if factorisable, use (px + q)(rx + s) = 0.
对于 ax² + bx + c = 0，若能分解，则写成 (px + q)(rx + s) = 0。
4. Solving by Completing the Square | 用配方法求解
Completing the square rewrites x² + bx in the form (x + p)² + q. This is useful when factorising is not easy. For example, x² + 6x + 2 = 0 can be rewritten as (x + 3)² – 9 + 2 = 0, so (x + 3)² = 7. Taking the square root gives x + 3 = ±√7, hence x = -3 ± √7.
配方法将 x² + bx 改写为 (x + p)² + q 的形式。当因式分解不容易时，这个方法很有用。例如，x² + 6x + 2 = 0 可以改写为 (x + 3)² – 9 + 2 = 0，即 (x + 3)² = 7。两边开平方得 x + 3 = ±√7，因此 x = -3 ± √7。
For a general quadratic ax² + bx + c = 0, divide by a first, then complete the square. The method is reliable and also reveals the vertex of the parabola.
对于一般的一元二次方程 ax² + bx + c = 0，先除以 a，再配方。这种方法很可靠，还能揭示抛物线的顶点。
x² + bx = (x + b/2)² – (b/2)²
x² + bx = (x + b/2)² – (b/2)²
5. Solving by Quadratic Formula | 用求根公式求解
The quadratic formula works for any quadratic equation and is particularly useful when the equation cannot be factorised. For ax² + bx + c = 0, the solutions are given by the formula below. You must memorise this formula because it is not provided in the Edexcel IGCSE formula sheet.
求根公式适用于任意一元二次方程，尤其当方程无法因式分解时。对于 ax² + bx + c = 0，解由下面的公式给出。你必须记住这个公式，因为 Edexcel IGCSE 公式表中并不提供它。
x = [-b ± √(b² – 4ac)] / (2a)
x = [-b ± √(b² – 4ac)] / (2a)
For example, solve 2x² – 3x – 2 = 0. Here a = 2, b = -3, c = -2. Substitute into the formula: x = [3 ± √(9 + 16)] / 4 = [3 ± 5] / 4. So x = 2 or x = -1/2.
例如，解 2x² – 3x – 2 = 0。这里 a = 2，b = -3，c = -2。代入公式：x = [3 ± √(9 + 16)] / 4 = [3 ± 5] / 4。所以 x = 2 或 x = -1/2。
6. The Discriminant | 判别式
The expression b² – 4ac inside the square root is called the discriminant, denoted Δ (Delta). It tells you the nature of the roots without solving the equation.
根号内的表达式 b² – 4ac 被称为判别式，记作 Δ（Delta）。不用解方程，它就能告诉你根的性质。
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If Δ > 0, there are two real and distinct roots.
若 Δ > 0，则有两个不同实数根。
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If Δ = 0, there is one repeated real root.
若 Δ = 0，则有一个重实数根。
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If Δ < 0, there are no real roots (two complex roots).
若 Δ < 0，则没有实数根（有两个复数根）。
For example, for x² – 4x + 4 = 0, Δ = 16 – 16 = 0, so the equation has exactly one root: x = 2.
例如，对于 x² – 4x + 4 = 0，Δ = 16 – 16 = 0，所以方程只有一个根：x = 2。
7. Graphs of Quadratic Functions | 二次函数的图像
The graph of y = ax² + bx + c is a parabola. If a > 0, the parabola opens upward; if a < 0, it opens downward. The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph.
函数 y = ax² + bx + c 的图像是抛物线。如果 a > 0，抛物线开口向上；如果 a < 0，抛物线开口向下。方程 ax² + bx + c = 0 的根就是图像与 x 轴的交点。
The vertex of the parabola can be found by completing the square. Its x-coordinate is given by x = -b/(2a). For example, y = x² – 6x + 5 has its vertex at x = 3, and since y(3) = -4, the vertex is (3, -4).
抛物线的顶点可以通过配方法求得。其 x 坐标为 x = -b/(2a)。例如，y = x² – 6x + 5 的顶点在 x = 3，且 y(3) = -4，所以顶点为 (3, -4)。
| Δ | Number of real roots | Graph shape | 实数根个数 | 图像形状 |
| Δ > 0 | 2 | crosses x-axis twice | 与 x 轴有两个交点 | |
| Δ = 0 | 1 | touches x-axis at one point | 与 x 轴相切于一点 | |
| Δ < 0 | 0 | does not touch x-axis | 不与 x 轴相交 |
8. Solving Word Problems | 解应用题
Quadratic equations often appear in geometric and area problems. For example, a rectangle has length (x + 4) cm and width (x – 1) cm, and its area is 36 cm². Write the area equation: (x + 4)(x – 1) = 36. Expanding gives x² + 3x – 4 = 36, so x² + 3x – 40 = 0. Factorise: (x + 8)(x – 5) = 0. Since x cannot be negative, x = 5.
一元二次方程经常出现在几何和面积问题中。例如，一个长方形的长为 (x + 4) cm，宽为 (x – 1) cm，面积为 36 cm²。可列出面积方程：(x + 4)(x – 1) = 36。展开得 x² + 3x – 4 = 36，即 x² + 3x – 40 = 0。因式分解得 (x + 8)(x – 5) = 0。由于 x 不能为负，所以 x = 5。
Always check that your answers make sense in the context of the problem. Negative lengths or times are usually invalid.
一定要检查你的答案在题目情境中是否有意义。负长度或负时间通常无效。
9. Common Mistakes | 常见错误
One common mistake is forgetting to set the equation equal to zero before factorising. Another is losing the ± sign when taking square roots. Also, when using the quadratic formula, students often misread the sign of b. For example, if b = -5, then -b should be +5.
一个常见错误是在因式分解前忘记把方程化为零的形式。另一个错误是开平方时丢掉 ± 号。另外，使用求根公式时，学生经常看错 b 的符号。例如，如果 b = -5，那么 -b 应为 +5。
Take care with fractions: x = 1/2 is not the same as x = 2. Always simplify your answers and leave them in exact form when required.
注意分数：x = 1/2 与 x = 2 不同。务必化简答案，并在需要时保留精确形式。
10. Choosing the Right Method | 选择合适的方法
In the exam, you often have a choice of method. Use factorising when the expression factorises easily. Use completing the square when you need the vertex or when the equation is already in a suitable form. Use the quadratic formula for any equation, especially when factorising is difficult or impossible.
在考试中，你通常可以自行选择方法。当表达式容易因式分解时，使用因式分解法。当需要求顶点或方程本身适合配方时，使用配方法。对于任何方程，尤其是难以分解或无法分解的方程，使用求根公式。
| Method | When to use | 方法 | 何时使用 |
| Factorising | Easy factorisation | 因式分解法 | 容易分解时 |
| Completing square | Finding vertex / exact surd roots | 配方法 | 求顶点或含根号精确根时 |
| Quadratic formula | Always works, use when not factorisable | 求根公式 | 总是可行，无法分解时使用 |
11. Practice Questions | 练习题目
Try these questions to test your understanding. (1) Solve x² – 7x + 10 = 0. (2) Solve 3x² + 5x – 2 = 0. (3) Show that the equation x² + 2x + 3 = 0 has no real roots. (4) Find the x-coordinates of the intersection of y = x² – 1 and y = 2x + 2.
尝试以下题目来检验你的理解。(1) 解 x² – 7x + 10 = 0。(2) 解 3x² + 5x – 2 = 0。(3) 说明方程 x² + 2x + 3 = 0 没有实数根。(4) 求 y = x² – 1 与 y = 2x + 2 交点的 x 坐标。
Answers: (1) x = 2 or x = 5. (2) x = 1/3 or x = -2. (3) Δ = 4 – 12 = -8 < 0. (4) Solve x² - 1 = 2x + 2 → x² - 2x - 3 = 0 → x = 3 or x = -1.
答案：(1) x = 2 或 x = 5。(2) x = 1/3 或 x = -2。(3) Δ = 4 – 12 = -8 < 0。(4) 解 x² - 1 = 2x + 2 → x² - 2x - 3 = 0 → x = 3 或 x = -1。
12. Summary | 总结
Quadratic equations are solved by factorising, completing the square, or using the quadratic formula. The discriminant tells you about the number of real roots. Graphs of quadratics are parabolas, and the roots are the x-intercepts. With regular practice, you will be able to identify the fastest method and avoid common traps.
一元二次方程可以通过因式分解、配方或求根公式来求解。判别式告诉我们实数根的个数。二次函数的图像是抛物线，根就是与 x 轴的交点。通过经常练习，你将能够迅速判断最快捷的方法，并避开常见陷阱。
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