📚 Quadratic Equations for IGCSE Mathematics | IGCSE 数学二次方程
A quadratic equation is one of the most important topics in the IGCSE Mathematics syllabus. It appears in algebra, graphs and problem-solving, and it forms the foundation for more advanced work in functions and calculus. This revision guide explains the standard form, three main solution methods, the discriminant, graph features and common exam applications.
二次方程是 IGCSE 数学大纲中最重要的主题之一。它出现在代数、图像和问题解决中，也是函数和微积分等更高级内容的基础。本复习指南将解释标准形式、三种主要解法、判别式、图像特征以及常见的考试应用。
1. What is a Quadratic Equation? | 什么是二次方程？
s 2. In IGCSE, you will usually see equations such as x² - 5x + 6 = 0 or 2x² + 3x - 2 = 0.
二次方程是一个二次多项式方程，也就是说变量（通常是 x）的最高次数是 2。在 IGCSE 中，你通常会看到诸如 x² - 5x + 6 = 0 或 2x² + 3x - 2 = 0 这样的方程。
Quadratic equations can have two real solutions, one repeated real solution, or no real solutions depending on the discriminant. They are different from linear equations, which have degree 1 and at most one solution.
根据判别式的不同，二次方程可以有两个实数解、一个重根实数解或没有实数解。它们与一次方程不同，一次方程的次数是 1，最多只有一个解。
2. Standard Form and Coefficients | 标准形式与系数
and a ≠ 0. The condition a ≠ 0 is essential because if a = 0, the equation becomes linear.
二次方程的标准形式是 ax² + bx + c = 0，其中 a、b 和 c 是常数，且 a ≠ 0。a ≠ 0 这个条件非常重要，因为如果 a = 0，方程就变成了一次方程。
- a is the coefficient of x², called the quadratic coefficient.
- b is the coefficient of x, called the linear coefficient.
- c is the constant term.
- a 是 x² 的系数，称为二次项系数。
- b 是 x 的系数，称为一次项系数。
- c 是常数项。
range the equation into standard form. For example, 3x² - 7 = 2x becomes 3x² - 2x - 7 = 0.
在求解之前，你应先把方程整理成标准形式。例如，3x² - 7 = 2x 应写成 3x² - 2x - 7 = 0。
3. Solving by Factorising | 因式分解法求解
two linear brackets. For example, x² - 5x + 6 = 0 can be factorised as (x - 2)(x - 3) = 0.
当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。例如，x² - 5x + 6 = 0 可以因式分解为 (x - 2)(x - 3) = 0。
3, you may need to split the middle term or use grouping: 2x² + 7x + 3 = (2x + 1)(x + 3).
对于 x² + bx + c 型的因式分解，需要找到两个数，它们的积为 c，和为 b。对于 2x² + 7x + 3，可能需要拆中项或使用分组法：2x² + 7x + 3 = (2x + 1)(x + 3)。
Always check your factorisation by expanding the brackets. If the expansion returns to the original expression, the factorisation is correct.
务必通过展开括号来检查因式分解是否正确。如果展开后回到原式，说明因式分解正确。
4. Zero Product Property | 零乘积性质
expressions must be zero. In symbols: if (x - p)(x - q) = 0, then x - p = 0 or x - q = 0.
零乘积性质指出：如果两个式子的乘积为零，那么至少有一个式子必须为零。用符号表示：如果 (x - p)(x - q) = 0，那么 x - p = 0 或 x - q = 0。
adratic as (x - 2)(x - 3) = 0, you solve x - 2 = 0 and x - 3 = 0 to obtain x = 2 or x = 3.
这个性质正是因式分解法如此有用的原因。将二次方程写成 (x - 2)(x - 3) = 0 后，求解 x - 2 = 0 和 x - 3 = 0，就得到 x = 2 或 x = 3。
You cannot apply this property unless the equation equals zero and the other side is fully factorised.
除非方程一边等于零，另一边已完全因式分解，否则不能使用这个性质。
5. Solving by Completing the Square | 配方法求解
Completing the square transforms a quadratic expression into the form (x + p)² + q. This is useful when the quadratic cannot be factorised easily and is also needed for finding the turning point of a quadratic graph.
配方法可以把二次式转化为 (x + p)² + q 的形式。当二次式不容易因式分解时，配方法非常有用，而且在求二次图像顶点时也需要用到。
+ 3)² - 8. Then solve (x + 3)² - 8 = 0 by adding 8, taking square roots, and isolating x.
对于 x² + 6x + 1 = 0，先写成 x² + 6x + 1 = (x + 3)² - 9 + 1 = (x + 3)² - 8。然后解 (x + 3)² - 8 = 0：加 8，开平方，再解出 x。
The same method works when a ≠ 1, but you should first divide the whole equation by a.
当 a ≠ 1 时，同样的方法仍然可行，但你需要先将整个方程除以 a。
6. The Quadratic Formula | 二次公式
specially helpful when factorising is difficult or when the coefficients are not integers.
二次公式可以直接给出 ax² + bx + c = 0 的解。当因式分解较困难或系数不是整数时，它特别有用。
x = (-b ± √b² - 4ac) ÷ 2a
To use the formula, identify a, b and c from the standard form, substitute them carefully, and simplify. Watch out for negative signs: if b is negative, -b becomes positive.
使用公式时，要从标准形式中确定 a、b 和 c，小心代入并化简。注意负号：如果 b 是负数，-b 就变成正数。
For 2x² - 3x - 5 = 0, a = 2, b = -3, c = -5. The solutions are x = (3 ± √9 + 40) ÷ 4 = (3 ± 7) ÷ 4, giving x = 2.5 or x = -1.
对于 2x² - 3x - 5 = 0，a = 2，b = -3，c = -5。解为 x = (3 ± √9 + 40)
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图