📚 Quadratic Equations and Functions | 二次方程与二次函数
Quadratic equations and functions form a cornerstone of IGCSE Mathematics. From solving simple equations to analysing parabolic motion, this topic appears in virtually every exam paper and provides the foundation for higher-level study.
二次方程与二次函数是IGCSE数学的核心内容。从求解简单方程到分析抛物线运动，这一主题几乎出现在每一份试卷中，并为更高层次的学习奠定基础。
1. Basic Form | 基本形式
A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The expression ax² + bx + c is called a quadratic expression.
二次方程是最高次数为2的多项式方程，其标准形式写作 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。表达式 ax² + bx + c 称为二次表达式。
re both quadratic equations. Note that the coefficient of x² must not be zero; otherwise, the equation becomes linear.
例如，2x² + 3x - 5 = 0 和 x² - 4 = 0 都是二次方程。注意 x² 的系数不能为零，否则方程就变成一次方程了。
2. Factorisation Method | 因式分解法
Solving quadratic equations by factorisation relies on the zero product property: if the product of two factors equals zero, then at least one of the factors must be zero.
用因式分解法解二次方程依赖于零积性质：如果两个因式的乘积等于零，那么至少有一个因式必须为零。
3) = 0. Then we set each factor to zero: x + 2 = 0 or x + 3 = 0, giving x = -2 or x = -3.
解 x² + 5x + 6 = 0，我们先进行因式分解：(x + 2)(x + 3) = 0。然后令每个因子为零：x + 2 = 0 或 x + 3 = 0，解得 x = -2 或 x = -3。
fference of two squares, or the sum/product method. For example, a² - b² = (a - b)(a + b).
始终检查二次式是否可以通过提取公因式、平方差公式或和积法进行因式分解。例如，a² - b² = (a - b)(a + b)。
3. Completing the Square | 配方法
Completing the square is a powerful technique that transforms a quadratic expression ax² + bx + c into the form a(x - h)² + k. This form makes it easy to identify the turning point of a parabola.
配方法是一种强大的技巧，它将二次表达式 ax² + bx + c 变换为 a(x - h)² + k 的形式。这种形式便于确定抛物线的顶点。
of 6 (which is 3), square it (giving 9), and rewrite: x² + 6x + 9 - 9 + 2 = (x + 3)² - 7.
以 x² + 6x + 2 为例，取 6 的一半是 3，平方得 9，然后改写：x² + 6x + 9 - 9 + 2 = (x + 3)² - 7。
ctor it out first. For 2x² - 8x + 5: 2(x² - 4x) + 5 = 2[(x - 2)² - 4] + 5 = 2(x - 2)² - 3.
当 x² 的系数不为 1 时，先提出该系数。对于 2x² - 8x + 5：2(x² - 4x) + 5 = 2[(x - 2)² - 4] + 5 = 2(x - 2)² - 3。
4. The Quadratic Formula | 求根公式
adratic formula can solve any quadratic equation ax² + bx + c = 0. The roots are given by:
求根公式可以求解任何二次方程 ax² + bx + c = 0。其根为：
x = (-b ± √b² - 4ac) / 2a
This formula is derived by completing the square on the general form. It works for all quadratics, including those that cannot be factorised with integers.
该公式通过对一般形式配方法推导而来。它适用于所有二次方程，包括那些无法用整数因式分解的方程。
Example: Solve 2x² - 4x - 6 = 0. Here a = 2, b = -4, c = -6. Substituting: x = (4 ± √16 + 48) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4. Thus x = 3 or x = -1.
例：解 2x² - 4x - 6 = 0。这里 a = 2，b = -4，c = -6。代入：x = (4 ± √16 + 48) / 4 = (4 ± √64) / 4 = (4 ± 8) / 4。因此 x = 3 或 x = -1。
5. The Discriminant | 判别式
he discriminant. It determines the nature of the roots without fully solving the equation.
Δ = b² - 4ac 这个表达式称为判别式。它无需完整解方程即可确定根的性质。
If Δ > 0, the equation has two distinct real roots.
若 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, the equation has one repeated real root (two equal roots).
若 Δ = 0，方程有一个重根（两个相等的实数根）。
If Δ < 0, the equation has no real roots (two complex roots).
若 Δ < 0，方程没有实数根（两个复数根）。
xample, for x² + 2x + 5 = 0, Δ = 4 - 20 = -16 < 0, so this equation has no real solutions.
例如，对于 x² + 2x + 5 = 0，Δ = 4 - 20 = -16 < 0，因此该方程没有实数解。
6. Vieta's Formulas | 根与系数的关系
then the sum and product of the roots satisfy simple relationships with the coefficients.
若 α 和 β 是 ax² + bx + c = 0 的根，则两根之和与两根之积与系数之间存在简单关系。
α + β = -b/a, αβ = c/a
These formulas allow us to construct a quadratic equation from given roots, or to find the value of symmetric expressions without computing the roots explicitly.
这些公式允许我们从给定的根构造二次方程，或者在不显式计算根的情况下求对称表达式的值。
uation whose roots are 3 and -2. Sum = 1, product = -6, so the equation is x² - x - 6 = 0.
例：求一个以 3 和 -2 为根的二次方程。两根之和 = 1，两根之积 = -6，因此方程为 x² - x - 6 = 0。
7. Graphs of Quadratic Functions | 二次函数图像
x² + bx + c is a parabola. When a > 0, the parabola opens upward (U-shaped); when a < 0, it opens downward (∩-shaped).
二次函数 y = ax² + bx + c 的图像是一条抛物线。当 a > 0 时，抛物线开口向上（U形）；当 a < 0 时，抛物线开口向下（∩形）。
The graph can intersect the x-axis at zero, one or two points, corresponding to the number of real roots of the quadratic equation.
图像与 x 轴的交点可以有零个、一个或两个，分别对应二次方程实数根的个数。
s of symmetry is x = -b/(2a); and the turning point (vertex) lies on the axis of symmetry.
图像的关键特征：y 截距为 c；x 截距为方程的根；对称轴为 x = -b/(2a)；顶点位于对称轴上。
8. Vertex Form | 顶点形式
as y = a(x - h)² + k is in vertex form, where (h, k) is the turning point of the parabola.
二次函数写成 y = a(x - h)² + k 即顶点形式，其中 (h, k) 是抛物线的顶点。
To convert from standard form to vertex form, use completing the square. This is particularly useful in optimisation problems where the maximum or minimum value is required.
从标准形式转换为顶点形式需要使用配方法。这在需要求最大值或最小值的优化问题中特别有用。
horizontally, k shifts it vertically, and a determines the vertical stretch and direction.
注意顶点形式还揭示了基本图像 y = x² 的变换：h 使图像水平平移，k 使图像垂直平移，a 决定纵向伸缩和开口方向。
9. Applications | 实际应用
Quadratic equations model many real-world scenarios, including projectile motion, area optimisation, and parabolic structures.
二次方程建模了许多现实世界场景，包括抛体运动、面积优化和抛物线结构。
g is the acceleration due to gravity, v₀ the initial velocity, and h₀ the initial height.
抛体运动：向上抛出的物体的高度 h 由 h(t) = -½gt² + v₀t + h₀ 给出，其中 g 是重力加速度，v₀ 是初速度，h₀ 是初始高度。
Area optimisation: to find the maximum area of a rectangle with a fixed perimeter, we set up a quadratic function of one variable and find the vertex.
面积优化：要在固定周长下求矩形最大面积，我们建立一个关于一元变量的二次函数并找到其顶点。
Revenue problems: profit and revenue functions often take parabolic forms, allowing businesses to find the optimal selling price.
收入问题：利润和收入函数通常呈抛物线形式，帮助企业找到最优售价。
10. Exam Tips | 考试技巧
In IGCSE examinations, always check whether the question requires exact answers or decimal approximations. State your answer in the requested form.
在IGCSE考试中，务必检查题目要求的是精确答案还是小数近似值。按要求形式写出答案。
Remember that a² - b² = (a - b)(a + b) and other basic factorisation patterns can simplify problems considerably.
记住 a² - b² = (a - b)(a + b) 等基本因式分解模式可以大大简化问题。
Read the question carefully: "solve by factorising" means you must show factorisation steps.
仔细阅读题目：要求"用因式分解法求解"时，你必须展示因式分解步骤。
When sketching a parabola, always label the intercepts and vertex.
画抛物线草图时，务必标出截距和顶点。
Check your solutions by substituting back into the original equation.
将解代回原方程检验答案。
Use the discriminant to check how many solutions to expect before solving.
求解前用判别式检查预期有几个解。
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