📚 Mastering Quadratics: Completing the Square, the Discriminant and Inequalities | 精通二次函数：配方法、判别式与不等式
Quadratic expressions and equations are central to the Cambridge AS & A Level Pure Mathematics 1 syllabus. A solid grasp of completing the square, the quadratic formula, the discriminant and quadratic inequalities will help you sketch parabolas, solve equations and tackle modelling problems with confidence.
二次函数表达式与方程是剑桥 AS 与 A Level 纯数学 1 课程的核心内容。扎实掌握配方法、二次公式、判别式以及二次不等式，将帮助你从容绘制抛物线、解方程并解决建模问题。
1. Standard Form and Key Features | 标准形式与关键特征
0, it opens downwards. The y-intercept is (0, c), and the line of symmetry is x = -b/(2a).
二次函数表达式可写为 ax² + bx + c，其中 a、b、c 为常数且 a ≠ 0。y = ax² + bx + c 的图像是一条抛物线。若 a > 0，抛物线开口向上；若 a < 0，则开口向下。y 轴截距为 (0, c)，对称轴为 x = -b/(2a)。
line x = -b/(2a) and substituting this value into the quadratic to find the y-coordinate.
当题目要求求转折点时，实际上就是求抛物线的顶点。你可以通过配方式找到顶点，也可以利用对称轴 x = -b/(2a)，并将该值代入二次式求出 y 坐标。
y = ax² + bx + c, a ≠ 0; x = -b/(2a)
The leading coefficient a controls the width and direction of the parabola. A large positive value of a gives a narrow upward curve, while a negative value reflects the graph in the x-axis. The constant c gives the height at which the curve crosses the y-axis.
首项系数 a 决定抛物线的宽窄与开口方向。a 为正且绝对值较大时，曲线向上且较窄；a 为负时，图像关于 x 轴翻折。常数项 c 表示曲线与 y 轴交点的高度。
2. Completing the Square | 配方法
x² + bx = (x + b/2)² - (b/2)². This process exposes the vertex and helps solve equations.
配方法将二次式改写为 a(x + p)² + q 的形式。对于首项系数为 1 的表达式 x² + bx，加上并减去 (b/2)²：x² + bx = (x + b/2)² - (b/2)²。这一过程能够显现顶点位置，并有助于解方程。
c - b²/(4a)). Do not memorise this as a formula only; practise the steps so you can reproduce it accurately.
对于一般形式 ax² + bx + c，先从首两项中提出 a，然后在括号内完成配方。结果为 y = a(x + b/(2a))² + (c - b²/(4a))。不要只背公式，更要练习每一步，以便能够准确还原。
ax² + bx + c = a(x + b/(2a))² + (c - b²/(4a))
For example, x² + 6x + 1 becomes (x + 3)² - 8. To check your answer, expand the completed-square form. If it does not return to the original expression, look for an error in the constant term.
例如，x² + 6x + 1 可化为 (x + 3)² - 8。检查答案时，可将配方式展开。如果无法还原为原表达式，请检查常数项是否有误。
plying back. Example: 2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² - 4] + 5 = 2(x + 2)² - 3.
当 a 不等于 1 时，先在括号内完成配方，再乘回。例如：2x² + 8x + 5 = 2(x² + 4x) + 5 = 2[(x + 2)² - 4] + 5 = 2(x + 2)² - 3。
3. Vertex Form and Graph Transformations | 顶点式与图像变换
y = x²: a horizontal shift, a vertical shift, and a stretch or reflection determined by a.
配方式 y = a(x - h)² + k 直接给出顶点 (h, k)。这一点非常有用，因为任何二次函数图像都可以看作 y = x² 经过平移、伸缩或翻折后的结果。
(3, 4). The positive 2 makes the parabola narrower and keeps it opening upwards. The inside term (x - 3) shifts the graph 3 units
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