📚 Solving Quadratic Equations for IGCSE | IGCSE 数学二次方程求解
Quadratic equations appear in almost every IGCSE Mathematics paper, from simple factorisation to applied problems involving areas, projectile motion, and optimised shapes. This article explains the three standard solution methods, the discriminant, graph interpretation, and common exam-style pitfalls.
二次方程几乎出现在每份 IGCSE 数学试卷中，从简单的因式分解到涉及面积、抛体运动和优化图形的应用题。本文讲解三种标准解法、判别式、图像解读以及常见的考试式易错点。
1. What is a Quadratic Equation? | 什么是二次方程
A quadratic equation in one variable is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were zero, the x² term would disappear and the equation would be linear.
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。如果 a 为零，x² 项就消失了，方程就变成一次方程。
The highest power of the unknown x is 2, which is why the equation has the name ‘quadratic’. In IGCSE, the unknown is usually written as x, but it may also appear as t, y, or another variable.
未知数 x 的最高次数是 2，这正是该方程被称为“二次”的原因。在 IGCSE 中，未知数通常写作 x，但也可能以 t、y 或其他变量出现。
Examples of quadratic equations include x² − 5x + 6 = 0, 2y² + 3y − 2 = 0, and t² − 9 = 0.
二次方程的例子包括 x² − 5x + 6 = 0、2y² + 3y − 2 = 0 和 t² − 9 = 0。
2. Standard Form and Coefficients | 标准形式与系数
Before solving, it is useful to write the equation in standard form ax² + bx + c = 0, where all terms are on one side and zero is on the other side. This makes the signs of coefficients clear and reduces sign errors.
求解之前，最好先把方程写成标准形式 ax² + bx + c = 0，即将所有项都移到一边，另一边为零。这样系数的符号就清晰可见，减少符号错误。
For example, 3x − 4 = 5x² is not in standard form. Rearranging gives 5x² − 3x + 4 = 0, so a = 5, b = −3 and c = 4.
例如，3x − 4 = 5x² 不是标准形式。重新整理为 5x² − 3x + 4 = 0，因此 a = 5、b = −3、c = 4。
If the equation contains fractions, you can multiply every term by the lowest common denominator. If it contains brackets, expand them first. These steps help you identify a, b and c accurately before applying any method.
如果方程含有分数，可以在每一项上乘以最小公分母。如果含有括号，就先展开。这些步骤帮助你在使用任何方法之前准确识别 a、b 和 c。
3. Solving by Factorisation | 因式分解法
Factorisation is often the fastest method when the quadratic can be written as a product of two linear factors. You need two numbers whose product equals ac and whose sum equals b.
当二次式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。你需要找到两个数，它们的乘积等于 ac，和等于 b。
For x² − 5x + 6 = 0, we need two numbers that multiply to 6 and add to −5. The numbers are −2 and −3, so (x − 2)(x − 3) = 0.
对于 x² − 5x + 6 = 0，我们需要两个数，它们相乘为 6，相加为 −5。这两个数是 −2 和 −3，因此 (x − 2)(x − 3) = 0。
Since a product is zero only if at least one factor is zero, set x − 2 = 0 or x − 3 = 0. Hence x = 2 or x = 3.
因为只有当至少一个因式为零时乘积才为零，所以令 x − 2 = 0 或 x − 3 = 0。因此 x = 2 或 x = 3。
For equations where a is not 1, such as 2x² + 3x − 2 = 0, use the grouping method: find two numbers multiplying to 2 × (−2) = −4 and adding to 3. The numbers are 4 and −1, then split the middle term: 2x² + 4x − x − 2 = 0 and factorise in pairs.
对于 a 不等于 1 的方程，如 2x² + 3x − 2 = 0，使用分组法：找到两个数，乘积为 2 × (−2) = −4，和为 3。这两个数是 4 和 −1，然后将中间项拆分：2x² + 4x − x − 2 = 0，再成对分解。
This gives 2x(x + 2) − 1(x + 2) = 0, so (2x − 1)(x + 2) = 0, and therefore x = 1/2 or x = −2.
这样得到 2x(x + 2) − 1(x + 2) = 0，所以 (2x − 1)(x + 2) = 0，因此 x = 1/2 或 x = −2。
4. Solving by Completing the Square | 配方法
Completing the square transforms ax² + bx + c into the form a(x + p)² + q, which is useful when factorisation is difficult and for finding the vertex of a parabola.
配方法把 ax² + bx + c 变成 a(x + p)² + q 的形式，当因式分解较困难以及求抛物线顶点时很有用。
For x² − 6x + 2 = 0, take half of −6 to get −3, square it to get 9, and rewrite as (x − 3)² − 9 + 2 = 0, so (x − 3)² = 7.
对于 x² − 6x + 2 = 0，取 −6 的一半得到 −3，平方得 9，重写为 (x − 3)² − 9 + 2 = 0，因此 (x − 3)² = 7。
Taking square roots gives x − 3 = ±√7, so x = 3 ± √7. These are exact solutions and do not need to be rounded unless the question requests it.
开平方得到 x − 3 = ±√7，所以 x = 3 ± √7。这些是精确解，除非题目要求，否则不需要四舍五入。
The expression (x − 3)² shows the vertex of y = x² − 6x + 2 is (3, −7). This link between algebra and graphs is frequently tested.
表达式 (x − 3)² 表明 y = x² − 6x + 2 的顶点是 (3, −7)。代数与图像之间的这种联系经常被考查。
5. The Quadratic Formula | 求根公式
The quadratic formula is a universal method for any quadratic equation ax² + bx + c = 0. It states:
求根公式是适用于任何二次方程 ax² + bx + c = 0 的通用方法。公式为：
x = (−b ± √(b² − 4ac)) / 2a
This formula is especially helpful when the quadratic does not factorise neatly, such as when the roots involve surds or decimals.
当二次式不能整齐地因式分解时，例如根含有根式或小数时，这个公式尤其有用。
For 2x² + 3x − 2 = 0, a = 2, b = 3 and c = −2. Substituting gives x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4.
对于 2x² + 3x − 2 = 0，a = 2、b = 3、c = −2。代入得 x = (−3 ± √(3² − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4。
The two solutions are x = (−3 + 5) / 4 = 1/2 and x = (−3 − 5) / 4 = −2.
两个解为 x = (−3 + 5) / 4 = 1/2 和 x = (−3 − 5) / 4 = −2。
If the equation is not in standard form, rearrange it first. Always quote the formula if you are using it in an exam, then substitute carefully.
如果方程不是标准形式，要先重新整理。在考试中使用该公式时，要先写出公式，然后仔细代入。
6. Discriminant and Nature of Roots |
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