📚 Mastering Quadratic Equations for IGCSE Mathematics | IGCSE 数学：掌握二次方程
Quadratic equations are one of the most important algebraic topics in IGCSE Mathematics. They appear in algebra, coordinate geometry, number problems, and real-world modelling questions. A quadratic equation can be written in the form ax² + bx + c = 0, where a ≠ 0. This article explains the main solution methods, the role of the discriminant, and key exam techniques step by step.
二次方程是 IGCSE 数学中最重要的代数主题之一。它们出现在代数、坐标几何、数值问题以及现实世界建模题中。二次方程可以写成 ax² + bx + c = 0 的形式，其中 a ≠ 0。本文将逐步讲解主要求解方法、判别式的作用以及关键考试技巧。
1. What is a Quadratic Equation? | 什么是二次方程？
A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable x is 2. A quadratic equation may have two real solutions, one repeated real solution, or no real solutions, depending on the value of the discriminant. The standard form is ax² + bx + c = 0, where a, b, and c are constants and a cannot be zero.
二次方程是一个次数为 2 的多项式方程。这意味着变量 x 的最高次数是 2。根据判别式的值，二次方程可能有两个实数解、一个重复实数解或没有实数解。其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数，且 a 不能为零。
Examples of quadratic equations include x² – 3x + 2 = 0, 2x² + 5x – 3 = 0, and 4x² – 9 = 0. Some equations may not look quadratic at first, such as 2x² + 5x = 8, but they can be rearranged into standard form by moving all terms to one side.
二次方程的例子包括 x² – 3x + 2 = 0、2x² + 5x – 3 = 0 和 4x² – 9 = 0。有些方程乍看可能不像二次方程，例如 2x² + 5x = 8，但把所有项移到一边后就可以化为标准形式。
2. Standard Form and Coefficients | 标准形式与系数
Before solving any quadratic equation, you should always rewrite it in the standard form ax² + bx + c = 0. This makes it easier to identify the coefficients a, b, and c correctly. Pay special attention to negative signs, because many exam mistakes come from using the wrong sign for b or c.
在求解任何二次方程之前，你都应该先将其改写为标准形式 ax² + bx + c = 0。这样可以更容易地正确识别系数 a、b 和 c。要特别注意负号，因为许多考试错误都源于 b 或 c 的符号使用错误。
For example, in the equation 3x² – x + 7 = 0, the coefficients are a = 3, b = -1, and c = 7. In the equation -2x² + 5 = 0, there is no x term, so b = 0. The coefficients are a = -2, b = 0, and c = 5. This careful identification is essential for using the quadratic formula later.
例如，在方程 3x² – x + 7 = 0 中，系数为 a = 3、b = -1、c = 7。在方程 -2x² + 5 = 0 中，没有 x 项，因此 b = 0。其系数为 a = -2、b = 0、c = 5。这种仔细的识别对以后使用求根公式非常重要。
3. Solving by Factorisation | 因式分解法
Factorisation is usually the fastest method when a quadratic equation has rational roots. The goal is to write the quadratic expression as a product of two linear brackets. Once the expression is factorised, each bracket can be set equal to zero and solved separately.
当二次方程有有理根时，因式分解通常是最快的方法。目标是把二次式写成两个一次括号的乘积。因式分解后，就可以令每个括号分别等于零并分别求解。
For example, to solve x² – 5x + 6 = 0, look for two numbers that multiply to 6 and add to -5. These numbers are -2 and -3. Therefore, x² – 5x + 6 = (x – 2)(x – 3). Setting each bracket to zero gives x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.
例如，要解 x² – 5x + 6 = 0，需找到两个数，使它们的乘积为 6，和为 -5。这两个数是 -2 和 -3。因此，x² – 5x + 6 = (x – 2)(x – 3)。令每个括号为零，得到 x – 2 = 0 或 x – 3 = 0，所以 x = 2 或 x = 3。
Use the following four steps when solving by factorisation:
使用因式分解法时，可以遵循以下四个步骤：
- Step 1: Write the equation in standard form. | 第一步：将方程写成标准形式。
- Step 2: Factorise the quadratic expression. | 第二步：对二次式进行因式分解。
- Step 3: Set each bracket equal to zero. | 第三步：令每个括号等于零。
- Step 4: Solve the resulting linear equations. | 第四步：解所得的一次方程。
4. Zero Product Property | 零乘积性质
The reason factorisation works is the zero product property. This property states that if AB = 0, then A = 0 or B = 0. It is only valid when the product equals zero, so you must bring all terms to one side before factorising. Dividing both sides by a variable is dangerous because it may remove a valid solution.
因式分解法之所以有效，是因为零乘积性质。该性质指出：如果 AB = 0，那么 A = 0 或 B = 0。只有当乘积等于零时才能使用这一性质，因此在进行因式分解前必须把所有项移到一边。两边同时除以一个变量是危险的，因为这可能会丢失一个有效解。
For example, consider x² = 4x. A common mistake is to divide both sides by x and obtain x = 4. However, this loses the solution x = 0. The correct approach is to write x² – 4x = 0, then factorise to get x(x – 4) = 0. The solutions are x = 0 or x = 4.
例如，考虑方程 x² = 4x。一个常见错误是两边同时除以 x，得到 x = 4。然而，这会丢失解 x = 0。正确的做法是写成 x² – 4x = 0，然后因式分解得到 x(x – 4) = 0。其解为 x = 0 或 x = 4。
5. Solving by Completing the Square | 配方法
Completing the square is a powerful method that rewrites a quadratic expression in the form (x + p)² + q. The key step is to take half of the coefficient of x, square it, then add and subtract that value. This method is especially useful for finding the vertex of a parabola and for deriving the quadratic formula.
配方法是一种强大的方法，它把二次式改写成 (x + p)² + q 的形式。关键步骤是取 x 项系数的一半并平方，然后加上再减去这个值。此方法对于求抛物线的顶点以及推导求根公式特别有用。
For example, to solve x² + 6x + 2 = 0, first take half of 6, which is 3, and square it to get 9. Then write x² + 6x + 2 = x² + 6x + 9 – 9 + 2 = (x + 3)² – 7. The equation becomes (x + 3)² – 7 = 0, so (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, and therefore x = -3 ± √7.
例如，要解 x² + 6x + 2 = 0，首先取 6 的一半，即 3，并将其平方得到 9。然后写 x² + 6x + 2 = x² + 6x + 9 – 9 + 2 = (x + 3)² – 7。方程变为 (x + 3)² – 7 = 0，所以 (x + 3)² = 7。两边开平方得到 x + 3 = ±√7，因此 x = -3 ± √7。
6. The Quadratic Formula | 求根公式
The quadratic formula is a universal method for solving any quadratic equation. For ax² + bx + c = 0, the solutions are given by the formula below. This method always works, even when factorisation is difficult or the roots are irrational or involve surds.
求根公式是求解任何二次方程的通用方法。对于 ax² + bx + c = 0，其解由以下公式给出。即使因式分解较难，或者根为无理数或含有根式，此方法也始终有效。
x = (-b ± √(b² – 4ac)) / (2a)
For example, solve 2x² + 3x – 2 = 0. Here a = 2, b = 3, and c = -2. Substituting into the formula gives x = (-3 ± √(3² – 4 × 2 × (-2))) / (2 × 2) = (-3 ± √(9 + 16)) / 4 = (-3 ± √25) / 4. This simplifies to x = (-3 + 5) / 4 = 1/2 or x = (-3 – 5) / 4 = -2.
例如，求解 2x² + 3x – 2 = 0。这里 a = 2、b = 3、c = -2。代入公式得到 x = (-3 ± √(3² – 4 × 2 × (-2))) / (2 × 2) = (-3 ± √(9 + 16)) / 4 = (-3 ± √25) / 4。化简后得到 x = (-3 + 5) / 4 = 1/2 或 x = (-3 – 5) / 4 = -2。
When you use the formula, always write the values of a, b, and c first. Use brackets around negative numbers when substituting, especially on a calculator, to avoid sign errors.
使用公式时，一定要先写出 a、b、c 的值。代入时，尤其是使用计算器时，应在负数周围加上括号，以避免符号错误。
7. The Discriminant | 判别式
The discriminant is the part of the quadratic formula under the square root: Δ = b² – 4ac. It tells you the nature of the roots without solving the equation fully. The discriminant is a very common IGCSE exam topic because it links algebra to the number of solutions or intersections.
判别式是求根公式中平方根下的部分：Δ = b² – 4ac。它可以在不解方程的情况下判断根的性质。判别式是 IGCSE 考试中非常常见的主题，因为它把代数与解的个数或交点数量联系起来。
The following table summarises the three cases:
下表总结了三种情况：
| Discriminant | Nature of roots | 根的性质 |
|---|---|---|
| Δ > 0 | two distinct real roots | 两个不同实根 |
| Δ = 0 | one repeated real root | 一个重复实根 |
| Δ < 0 | no real roots | 没有实根 |
For example, in x² – 6x + 9 = 0, the discriminant is (-6)² – 4 × 1 × 9 = 36 – 36 = 0, so the equation has one repeated root, which is x = 3.
例如，在 x² – 6x + 9 = 0 中，判别式为 (-6)² – 4 × 1 × 9 = 36 – 36 = 0，因此该方程有一个重复根，即 x = 3。
8. Graphical Interpretation | 图像意义
The graph of y = ax² + bx + c is a parabola. If a is positive, the parabola opens upwards and has a minimum point. If a is negative, it opens downwards and has a maximum point. The solutions of ax² + bx + c = 0 are the x-coordinates where the
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