📚 Quadratic Equations for IGCSE Maths: Solving, Graphs and Applications | IGCSE 数学二次方程：求解、图像与应用
Quadratic equations are one of the most important topics in IGCSE Mathematics. They appear in algebra, coordinate geometry and word problems, and they provide a foundation for more advanced study.
二次方程是 IGCSE 数学最重要的主题之一。它们出现在代数、坐标系几何和文字题中，并且为更深入的学习打下基础。
1. What is a Quadratic Equation? | 什么是二次方程
er of the variable, usually x, is 2. For example, x² + 5x + 6 = 0 is a quadratic equation.
二次方程是次数为 2 的多项式方程。变量（通常为 x）的最高次数是 2。例如，x² + 5x + 6 = 0 就是一个二次方程。
In IGCSE, we normally work with one variable and real coefficients. The standard form is shown below.
在 IGCSE 中，我们通常处理一个变量和实系数。标准形式如下所示。
ax² + bx + c = 0, a ≠ 0
Here a, b and c are constants, a is the coefficient of x², b is the coefficient of x, and c is the constant term.
这里 a、b 和 c 是常数，a 是 x² 的系数，b 是 x 的系数，c 是常数项。
equation is no longer quadratic; it becomes a linear equation. This is a common exam trap.
如果 a = 0，方程就不再是二次方程，而变成一次方程。这是考试中常见的陷阱。
2. Standard Form and Key Terms | 标准形式与关键术语
= 0. This means collecting all terms on one side and setting the other side equal to zero.
在求解任何二次方程之前，你必须先把它整理成标准形式 ax² + bx + c = 0。这意味着把所有项移到一边，并让另一边等于 0。
For example, 3x² = 7 - 2x should be rewritten as 3x² + 2x - 7 = 0.
例如，3x² = 7 - 2x 应改写为 3x² + 2x - 7 = 0。
The solutions of a quadratic equation are also called roots or x-intercepts when the equation is linked to a graph.
二次方程的解也称为根，或在与图像联系时称为 x 轴截距。
Key terms include: quadratic term (ax²), linear term (bx), constant term (c), leading coefficient (a), roots, and factors.
关键术语包括：二次项（ax²）、一次项（bx）、常数项（c）、首项系数（a）、根和因式。
3. Solving by Factorising | 因式分解法求解
Factorising is often the fastest method when the quadratic can be written as a product of two linear brackets. The equation must be in standard form first.
当二次方程可以写成两个一次括号的乘积时，因式分解法通常是最快的方法。方程必须先化为标准形式。
roperty: if p × q = 0, then p = 0 or q = 0. This is why we set each bracket equal to zero.
我们使用零乘积性质：如果 p × q = 0，那么 p = 0 或 q = 0。这就是为什么我们把每个括号设为零。
rise to get (x + 2)(x + 3) = 0. Therefore x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.
例如：解 x² + 5x + 6 = 0。因式分解得到 (x + 2)(x + 3) = 0。因此 x + 2 = 0 或 x + 3 = 0，得到 x = -2 或 x = -3。
x² + 5x + 6 = 0 ⇒ (x + 2)(x + 3) = 0 ⇒ x = -2 or x = -3
Always check your factorisation by expanding the brackets. This prevents sign errors.
务必通过展开括号来检查因式分解。这可以防止符号错误。
4. Solving by Completing the Square | 配方法求解
Completing the square is a reliable method that works for any quadratic equation. It is especially useful when the quadratic cannot be factorised easily.
配方法是一种适用于任何二次方程的可靠方法。当二次方程不易因式分解时，它尤其有用。
0, the idea is to write x² + bx as (x + b/2)² - (b/2)². Then bring the constant term back.
对于 x² + bx + c = 0，其思路是把 x² + bx 写成 (x + b/2)² - (b/2)²。然后再把常数项放回来。
Example: x² + 6x + 5 = 0 becomes (x + 3)² - 9 + 5 = 0, so (x + 3)² - 4 = 0.
例如：x² + 6x + 5 = 0 可写成 (x + 3)² - 9 + 5 = 0，即 (x + 3)² - 4 = 0。
Then (x + 3)² = 4, so x + 3 = ±√4, giving x = -3 ± 2, hence x = -1 or x = -5.
然后 (x + 3)² = 4，所以 x + 3 = ±√4，得到 x = -3 ± 2，因此 x = -1 或 x = -5。
x² + 6x + 5 = 0 ⇒ (x + 3)² - 4 = 0 ⇒ x = -1 or x = -5
When the leading coefficient a is not 1, factor it out from the first two terms before completing the square.
当首项系数 a 不为 1 时，在配方前要先把前两项中的 a 提出来。
5. The Quadratic
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