Quadratic Functions and Equations | 二次函数与方程

📚 Quadratic Functions and Equations | 二次函数与方程

Quadratic functions and equations form one of the most important topics in the IGCSE Mathematics syllabus. From factorisation to the quadratic formula, this topic tests your algebraic manipulation, graphical interpretation, and problem-solving skills. In this revision guide, we will break down everything you need to know into clear, exam-focused sections.

二次函数与方程是IGCSE数学大纲中最重要的主题之一。从因式分解到求根公式，这一主题考查你的代数运算、图像解读和问题解决能力。在本复习指南中，我们将把所需掌握的全部内容拆解为清晰的、紧扣考点的章节。

1. The General Form of a Quadratic Function | 二次函数的一般形式

A quadratic function is a polynomial of degree 2. Its standard form is written as:

二次函数是次数为2的多项式，其标准形式写作：

f(x) = ax² + bx + c, where a ≠ 0

Here, \(a\), \(b\), and \(c\) are constants, and \(a\) must not be zero. If \(a = 0\), the function becomes linear. The graph of a quadratic function is a curve called a parabola.

其中 \(a\)、\(b\)、\(c\) 为常数，且 \(a\) 不能为零。如果 \(a = 0\)，函数就变成线性函数。二次函数的图像是一条称为抛物线的曲线。

If a > 0, the parabola opens upward (U-shaped). 若 a > 0，抛物线开口朝上（U形）。
If a < 0, the parabola opens downward (∩-shaped). 若 a < 0，抛物线开口朝下（∩形）。

The sign of \(a\) determines not only the direction of the curve but also whether the function has a maximum or a minimum value.

\(a\) 的符号不仅决定曲线的开口方向，还决定函数具有最大值还是最小值。

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

Factorisation is the fastest method when the quadratic expression can be factorised easily. The principle is based on the zero product property: if the product of two numbers is zero, then at least one of them must be zero.

当二次表达式可以轻松因式分解时，因式分解是最快的方法。其原理基于零乘积性质：如果两个数的乘积为零，则至少其中一个数必须为零。

Example: Solve x² − 5x + 6 = 0.

示例：解方程 x² − 5x + 6 = 0。

(x − 2)(x − 3) = 0

x − 2 = 0 或 x − 3 = 0

x = 2 或 x = 3

When the coefficient of \(x^2\) is not 1, you may need to factorise by grouping or use the “cross method”. Always expand your answer to check your work.

当 \(x^2\) 的系数不是1时，你可能需要使用分组分解法或十字相乘法。始终通过展开你的答案来检查计算是否正确。

3. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula provides a universal solution. For the equation ax² + bx + c = 0:

当因式分解困难或无法进行时，求根公式提供了一种通用的解法。对于方程 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) / (2a)

This formula is derived by completing the square and works for all quadratic equations, including those with irrational roots.

该公式通过配方法推导而来，适用于所有二次方程，包括具有无理数根的方程。

Example: Solve 2x² + 3x − 1 = 0.

示例：解方程 2x² + 3x − 1 = 0。

Here a = 2, b = 3, c = −1. Substituting:

其中 a = 2，b = 3，c = −1。代入得：

x = (−3 ± √(9 − 4 × 2 × (−1))) / (2 × 2) = (−3 ± √17) / 4

So the two solutions are x ≈ 0.28 and x ≈ −1.78 (to 2 decimal places).

因此两个解约为 x ≈ 0.28 和 x ≈ −1.78（精确到两位小数）。

4. The Discriminant and the Nature of Roots | 判别式与根的性质

The expression \(b^2 − 4ac\) is called the discriminant. It tells us how many and what type of roots a quadratic equation has without actually solving it.

表达式 \(b^2 − 4ac\) 称为判别式。它能在实际解方程之前告诉我们二次方程有多少个根以及根的类型。

判别式 Δ	根的性质
Δ > 0	两个不相等的实数根
Δ = 0	一个重根（两个相等的实数根）
Δ < 0	无实数根（两个复数根）

The discriminant also tells us the number of x-intercepts of the parabola: two distinct points for Δ > 0, one point for Δ = 0, and no x-intercepts for Δ < 0.

判别式还告诉我们抛物线与x轴的交点数量：Δ > 0 时有两个不同的交点，Δ = 0 时有一个交点，Δ < 0 时没有交点。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x − h)² + k, which directly reveals the vertex of the parabola. This is a vital skill for both solving equations and graphing.

配方法将二次式改写为 a(x − h)² + k 的形式，直接揭示抛物线的顶点。这是解方程和绘制图像的关键技能。

Example: Express x² + 6x + 5 in the form (x + p)² + q.

示例：将 x² + 6x + 5 写成 (x + p)² + q 的形式。

Take half of 6, which is 3, and write:

取6的一半，即3，然后写：

x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4

The vertex is at (−3, −4). In general, for ax² + bx + c completed as a(x − h)² + k, the vertex is at (h, k).

顶点在 (−3, −4)。一般来说，将 ax² + bx + c 配方为 a(x − h)² + k 后，顶点在 (h, k)。

6. Graphs of Quadratic Functions | 二次函数的图像

To sketch a quadratic graph accurately, you need to identify key features: the vertex (turning point), the y-intercept, the x-intercepts (if any), and the axis of symmetry.

要准确绘制二次函数图像，你需要确定关键特征：顶点（转折点）、y轴截距、x轴截距（如有）以及对称轴。

y-intercept: set x = 0, giving y = c. y轴截距：令 x = 0，得 y = c。
x-intercepts: solve f(x) = 0. x轴截距：解方程 f(x) = 0。
Axis of symmetry: x = −b / (2a). 对称轴：x = −b / (2a)。
Vertex: x-coordinate is −b/(2a); y-coordinate is f(−b/(2a)). 顶点：横坐标为 −b/(2a)，纵坐标为 f(−b/(2a))。

For example, for f(x) = x² − 4x + 3, the axis of symmetry is x = 2, the vertex is (2, −1), the y-intercept is (0, 3), and the roots are x = 1 and x = 3.

例如，对于 f(x) = x² − 4x + 3，对称轴为 x = 2，顶点为 (2, −1)，y轴截距为 (0, 3)，根为 x = 1 和 x = 3。

7. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities involves finding the range of x values for which the quadratic expression is positive or negative. This is typically done by first finding the roots and then testing intervals.

解二次不等式需要找到使二次表达式为正或负的x值范围。通常先求根，然后测试区间。

Example: Solve x² − x − 2 > 0.

示例：解不等式 x² − x − 2 > 0。

First factorise: (x − 2)(x + 1) > 0. The roots are x = 2 and x = −1. The parabola opens upward, so the expression is positive outside the roots:

首先因式分解：(x − 2)(x + 1) > 0。根为 x = 2 和 x = −1。抛物线开口向上，因此表达式在两根之外为正：

x < −1 或 x > 2

Always sketch a quick graph and test a value in each interval to confirm your answer.

始终画一个快速草图并在每个区间内测试一个值来确认你的答案。

8. Word Problems Involving Quadratics | 二次方程应用题

Many exam questions require you to translate a real-world situation into a quadratic equation. These often involve area, projectile motion, or number relationships. The strategy is to define a variable, form an equation, solve it, and then interpret the answer in context.

许多考试题目要求你将现实情境转化为二次方程。这些通常涉及面积、抛体运动或数字关系。解题策略是设变量、列方程、求解，然后在情境中解释答案。

Example: A rectangle has length (x + 3) cm and width x cm. Its area is 28 cm². Find x.

示例：一个矩形的长为 (x + 3) cm，宽为 x cm，面积为 28 cm²。求 x。

x(x + 3) = 28 → x² + 3x − 28 = 0 → (x + 7)(x − 4) = 0

So x = 4 (reject x = −7 since length cannot be negative).

因此 x = 4（舍去 x = −7，因为长度不能为负）。

9. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, the following useful relationships hold:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，以下有用关系成立：

α + β = −b / a

α × β = c / a

These relationships allow you to write a quadratic equation given its roots. For example, if the roots are 2 and −5, then:

这些关系允许你在已知根的情况下写出二次方程。例如，如果根为 2 和 −5，则：

x² − (2 + (−5))x + (2 × (−5)) = x² + 3x − 10 = 0

This technique is especially useful for “forming a quadratic equation” questions in the non-calculator paper.

这一技巧在非计算器试卷中”构造二次方程”的题目中尤为有用。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Below are the most common pitfalls students fall into, along with practical advice to avoid them in the exam.

以下是学生最常掉入的陷阱，以及考试中避免这些错误的实用建议。

Forgetting that a ≠ 0. 忘记条件 a ≠ 0。
Misapplying the quadratic formula signs — be careful with negative b. 用错求根公式中的符号——注意负的 b。
Dividing both sides by x when x may be zero. 在 x 可能为零时两边同除以 x。
Forgetting to set the equation to zero before solving. 求解前忘记将方程化为零。
Not checking answers by substitution. 不通过代入检验答案。

Always read the question carefully: does it ask for roots, coordinates of the vertex, or the range for an inequality? These require different methods.

始终仔细审题：题目要求根、顶点坐标，还是不等式的范围？这些需要不同的方法。

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Quadratic Functions and Equations | 二次函数与方程