📚 Quadratic Equations | 二次方程
A quadratic equation is one of the central topics in A-Level Pure Mathematics. It is expressed in a standard form and solved by a variety of algebraic techniques, all of which the AQA examination may test directly or through hidden application.
二次方程是 A-Level 纯数学中的核心内容。它以标准形式呈现，并通过多种代数方法求解；在 AQA 考试中，这些知识点既可能直接考察，也可能以隐含形式出现在应用题中。
1. Standard Form and Solutions | 标准形式与解
A quadratic equation is an equation of the form:
二次方程是指具有如下形式的方程：
ax² + bx + c = 0, a ≠ 0
Here a, b and c are real numbers, and a is the coefficient of x². The values of x that make the equation true are called the roots or solutions.
其中 a、b、c 是实数，a 是 x² 的系数。使方程成立的 x 值称为根或解。
A quadratic equation can have two distinct real roots, one repeated real root, or no real roots, depending on the constant values involved. This is determined by the discriminant, which is covered in later sections.
根据所含常数的不同，二次方程可能有两个不相等的实数根、一个重根或没有实数根。这一性质由后续将介绍的判别式决定。
2. Solving by Factorisation | 因式分解法
Factorisation is often the quickest method when the quadratic expression can be written as a product of two linear factors.
当二次式能够写成两个一次因式的乘积时，因式分解通常是最快捷的方法。
² - 5x + 6 = 0. We seek two numbers that multiply to 6 and add to -5; these are -2 and -3.
例如，解 x² - 5x + 6 = 0。我们寻找两个数，它们相乘得 6、相加得 -5；这两个数是 -2 和 -3。
(x - 2)(x - 3) = 0
oduct of real numbers is zero then at least one factor is zero, so x - 2 = 0 or x - 3 = 0.
根据零积性质，若实数的乘积为零，则至少有一个因数为零，所以 x - 2 = 0 或 x - 3 = 0。
x = 2 or x = 3
For quadratics of the form x² + bx + c, always look for two numbers whose product is c and sum is b. If the coefficient of x² is not 1, use trial factors or the multiplication method such as factorising the pair grouping.
对于形如 x² + bx + c 的二次方程，总是寻找乘积为 c、和为 b 的两个数。如果 x² 的系数不是 1，可以使用试因法，或者通过分组成对来分解。
3. Solving by Completing the Square | 配方法
Completing the square rewrites a quadratic expression in the form a(x + p)² + q. This is useful for solving equations and for finding the vertex of a parabola.
配方法将二次式改写成 a(x + p)² + q 的形式。这种技巧既可用于求解方程，也可用于寻找抛物线的顶点。
Consider x² + 6x + 2 = 0. The first two terms, x² + 6x, can be written as
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