Quadratic Equations and Functions | 二次方程与二次函数

📚 Quadratic Equations and Functions | 二次方程与二次函数

Quadratic equations and functions form a central part of the IGCSE Mathematics syllabus. Mastering their algebraic and graphical techniques is essential for achieving high grades.

二次方程与二次函数是 IGCSE 数学课程的核心内容。掌握它们的代数与图形技巧对取得高分至关重要。

1. Standard Forms of Quadratic Expressions | 二次表达式的标准形式

A quadratic expression in one variable x can be written in three useful forms. The general form is ax² + bx + c = 0, where a ≠ 0. The factored form is a(x − p)(x − q), and the completed square form is a(x − h)² + k.

一元二次表达式有三种常用形式。一般形式为 ax² + bx + c = 0，其中 a ≠ 0；因式分解形式为 a(x − p)(x − q)；配方法形式为 a(x − h)² + k。

The coefficient a determines the direction of the parabola: if a > 0 the graph opens upward, and if a < 0 it opens downward.

系数 a 决定抛物线的开口方向：a > 0 时开口向上，a < 0 时开口向下。

2. Solving Quadratic Equations by Factorisation | 用因式分解解二次方程

When a quadratic equation can be factorised, this is usually the fastest solution method. For ax² + bx + c = 0, find two numbers whose product is ac and whose sum is b. Then split the middle term and factorise by grouping.

当二次方程可以因式分解时，这通常是最快的解法。对于 ax² + bx + c = 0，找两个数，其乘积为 ac，和为 b。然后用拆中项并分组分解的方法。

For example:

例如：

x² − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0 ⇒ x = 2 or x = 3.

Always check that the equation is set equal to zero before factorising.

分解前务必确保方程右边为零。

3. Solving by Completing the Square | 配方法

Completing the square rewrites x² + bx as (x + b/2)² − (b/2)². This method works for any quadratic, even when it does not factorise easily.

配方法将 x² + bx 改写为 (x + b/2)² − (b/2)²。这一方法适用于任何二次方程，即使不易因式分解。

For a quadratic in the form x² + bx + c = 0, add (b/2)² and then rearrange:

对于形如 x² + bx + c = 0 的方程，先加 (b/2)²，再整理：

(x + b/2)² = (b/2)² − c ⇒ x = −b/2 ± √((b/2)² − c).

This process also reveals the coordinates of the turning point directly.

该过程也能直接揭示顶点的坐标。

4. Solving by Quadratic Formula | 求根公式

The quadratic formula is derived from completing the square. For ax² + bx + c = 0, the solutions are given by:

二次求根公式由配方法推导得出。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) / (2a).

Substitute the coefficients a, b and c into the formula. Be careful with negative signs when substituting.

将系数 a、b、c 代入公式。代入负数时要特别注意符号。

5. The Discriminant | 判别式

The expression Δ = b² − 4ac is called the discriminant. It determines the nature of the roots.

表达式 Δ = b² − 4ac 称为判别式。它决定根的性质。

If Δ > 0, there are two distinct real roots.

若 Δ > 0，则有两个不同的实数根。
If Δ = 0, there is exactly one real root (a repeated root).

若 Δ = 0，则有一个实数根（重根）。
If Δ < 0, there are no real roots; the roots are complex.

若 Δ < 0，则没有实数根；根为复数。

The discriminant also tells us whether the graph crosses, touches, or misses the x-axis.

判别式还能告诉我们图像与 x 轴相交、相切还是相离。

6. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola with a vertical axis of symmetry. The axis of symmetry has equation x = −b/(2a).

y = ax² + bx + c 的图像是抛物线，对称轴为直线 x = −b/(2a)。

The y-intercept is at (0, c). The x-intercepts, when they exist, are the real roots of the equation.

y 截距为 (0, c)。x 截距（若存在）就是方程 ax² + bx + c = 0 的实数根。

Sketching a quadratic requires identifying the direction of opening, the y-intercept, the roots, and the turning point.

画出二次函数草图需要确定开口方向、y 截距、根以及顶点。

7. Finding the Turning Point | 求顶点

The turning point (vertex) of a quadratic graph can be found by completing the square. For y = a(x − h)² + k, the vertex is at (h, k).

二次函数图像的顶点可以通过配方找到。对于 y = a(x − h)² + k，顶点为 (h, k)。

Alternatively, the x-coordinate of the vertex is x = −b/(2a). Substitute this value into the equation to find the y-coordinate.

另一种方法：顶点的 x 坐标为 x = −b/(2a)。代入原式即可求出 y 坐标。

The vertex represents a minimum if a > 0 and a maximum if a < 0.

当 a > 0 时顶点为最小值点；当 a < 0 时顶点为最大值点。

8. Roots and Coefficients | 根与系数的关系

If the roots of ax² + bx + c = 0 are α and β, then the sum of the roots is α + β = −b/a and the product of roots is αβ = c/a.

若 ax² + bx + c = 0 的根为 α 和 β，则根之和为 α + β = −b/a，根之积为 αβ = c/a。

These relations are useful for solving problems without explicitly finding the roots, such as finding the value of α² + β².

这些关系可用于不解根而直接解决问题，例如求 α² + β² 的值。

Note: α² + β² = (α + β)² − 2αβ.

注意：α² + β² = (α + β)² − 2αβ。

9. Modelling with Quadratics | 二次函数建模

Many real-world problems, such as projectile motion, area optimisation and revenue calculations, can be modelled by quadratic functions.

许多实际问题，例如抛体运动、面积优化和收入计算，都可以用二次函数建模。

When setting up a model, define the variable clearly, write the quadratic expression, and interpret the roots or turning point in the context of the problem.

建立模型时，应清晰定义变量，写出二次表达式，并结合实际问题解释根或顶点的意义。

For example, the maximum height of a projectile occurs at the vertex of the height-time graph.

例如，抛体的最大高度出现在高度-时间图像的顶点处。

10. Common Exam Pitfalls | 常见考试陷阱

The most frequent errors in quadratic questions include forgetting to set the equation to zero before factorising, misapplying the quadratic formula with negative coefficients, and losing solutions when taking square roots.

二次函数题中最常见的错误包括：分解前忘记把方程化为零；对负数系数套用求根公式出错；开平方时漏掉一个解。

Always write the expression in standard form before solving.

解题前务必把表达式写成标准形式。
When solving x² = k, write x = ±√k, not just x = √k.

解 x² = k 时，应写成 x = ±√k，而不仅是 x = √k。
Check if the answer can be substituted back into the original equation.

检查答案是否可以代回原方程验证。

Practising past-paper style questions will help you avoid these pitfalls and build confidence.

练习历年真题风格的问题有助于避免这些陷阱并增强信心。

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Quadratic Equations and Functions | 二次方程与二次函数