Quadratic Equations and Functions | 二次方程与函数

📚 Quadratic Equations and Functions | 二次方程与函数

Quadratic equations and functions form one of the most essential topics in the IGCSE Mathematics syllabus. Mastery of this area unlocks success in algebra, graphing, and problem-solving across the entire examination.

二次方程与函数是 IGCSE 数学大纲中最核心的内容之一。熟练掌握这一领域，是你在代数、图像绘制以及全卷解题中获得成功的关键。

---

1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is any equation that can be written in the standard form:

任何可以写成以下标准形式的方程即为二次方程：

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are real numbers, with a being the coefficient of x², b the coefficient of x, and c the constant term. The condition a ≠ 0 is crucial — if a = 0, the equation becomes linear, not quadratic.

其中 a、b、c 为实数，a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 至关重要——若 a = 0，方程就变为一次方程，而非二次方程。

For example, 2x² − 5x + 3 = 0 is a quadratic equation with a = 2, b = −5, and c = 3.

例如，2x² − 5x + 3 = 0 就是一个二次方程，其中 a = 2，b = −5，c = 3。

---

2. The Graph of a Quadratic Function | 二次函数的图像

The graph of a quadratic function y = ax² + bx + c is a smooth, symmetric curve called a parabola. The shape of the parabola depends on the sign of a.

二次函数 y = ax² + bx + c 的图像是一条平滑且对称的曲线，称为抛物线。抛物线的开口方向取决于 a 的符号。

• If a > 0, the parabola opens upward (U-shape) and has a minimum point. | 若 a > 0，抛物线开口向上（呈 U 形），具有最小值点。
• If a < 0, the parabola opens downward (∩-shape) and has a maximum point. | 若 a < 0，抛物线开口向下（呈 ∩ 形），具有最大值点。

The turning point (vertex) of the parabola can be found using the formula:

抛物线的顶点（转向点）可以通过以下公式求得：

x = −b ÷ (2a)

Substitute this value of x back into the equation to find the corresponding y-coordinate of the vertex.

将此 x 值代回原方程，即可求得顶点对应的 y 坐标。

---

3. The Axis of Symmetry | 对称轴

Every parabola has a vertical line of symmetry that passes through its vertex. This line is called the axis of symmetry, and its equation is:

每条抛物线都有一条经过顶点的竖直对称线，称为对称轴，其方程为：

x = −b ÷ (2a)

For example, for the function y = x² − 4x + 5, the axis of symmetry is x = −(−4) ÷ (2 × 1) = 2. This means the parabola is a mirror image about the vertical line x = 2.

例如，对于函数 y = x² − 4x + 5，对称轴为 x = −(−4) ÷ (2 × 1) = 2。这意味着抛物线关于竖直线 x = 2 成镜像对称。

Understanding the axis of symmetry helps you sketch the graph quickly and locate the vertex accurately.

理解对称轴可以帮助你快速画出图像，并准确定位顶点。

---

4. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method for solving quadratic equations when the expression factorises neatly. The principle is based on the zero product property: if A × B = 0, then A = 0 or B = 0.

当二次表达式能够整齐地分解时，因式分解法往往是最快捷的解法。其原理基于零乘积性质：若 A × B = 0，则 A = 0 或 B = 0。

Consider the equation x² − 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give −5. These numbers are −2 and −3.

考虑方程 x² − 5x + 6 = 0。我们需要找到两个数，它们相乘得 6，相加得 −5。这两个数是 −2 和 −3。

(x − 2)(x − 3) = 0

x = 2 or x = 3

Always rearrange the equation into the form ax² + bx + c = 0 before attempting to factorise.

在尝试因式分解之前，务必先将方程整理为 ax² + bx + c = 0 的形式。

---

5. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula provides a universal solution. For any quadratic 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。对于任意二次方程 ax² + bx + c = 0：

x = (−b ± √(b² − 4ac)) ÷ (2a)

The symbol ± means that we perform the calculation twice: once with addition and once with subtraction, yielding two solutions.

符号 ± 表示我们需要进行两次计算：一次用加号，一次用减号，从而得到两个解。

For example, solve 2x² + 3x − 2 = 0 using the formula:

例如，用公式求解 2x² + 3x − 2 = 0：

x = (−3 ± √(9 + 16)) ÷ 4 = (−3 ± 5) ÷ 4

Thus, x = (−3 + 5) ÷ 4 = 0.5 or x = (−3 − 5) ÷ 4 = −2.

因此，x = (−3 + 5) ÷ 4 = 0.5 或 x = (−3 − 5) ÷ 4 = −2。

---

6. Completing the Square | 配方法

Completing the square is a powerful technique that rewrites a quadratic expression in the form a(x + p)² + q. This form reveals the vertex of the parabola directly.

配方法是一种强大的技巧，它可以将二次表达式改写为 a(x + p)² + q 的形式。这种形式可以直接揭示抛物线的顶点。

To complete the square for x² + 6x + 5:

对 x² + 6x + 5 配方：

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

The general procedure is: take half of the coefficient of x, square it, add and subtract this value, then simplify.

一般步骤是：取 x 系数的一半，将其平方，然后加一项并减一项，最后化简。

From the completed square form, the vertex of y = (x + 3)² − 4 is at (−3, −4).

从配方的形式中可知，y = (x + 3)² − 4 的顶点在 (−3, −4)。

---

7. The Discriminant | 判别式

The expression b² − 4ac inside the quadratic formula is called the discriminant, denoted by Δ. It determines the nature of the roots without solving the equation.

求根公式中的表达式 b² − 4ac 称为判别式，记作 Δ。它可以不求解方程而直接判断根的性质。

The three cases are as follows:

分以下三种情况：

Discriminant | 判别式	Nature of Roots | 根的性质
Δ > 0	Two distinct real roots | 两个不相等的实数根
Δ = 0	One repeated real root | 两个相等的实数根（重根）
Δ < 0	No real roots | 没有实数根

For example, for x² − 4x + 4 = 0, Δ = 16 − 16 = 0, so there is exactly one real root: x = 2.

例如，对于 x² − 4x + 4 = 0，Δ = 16 − 16 = 0，所以只有一个实数根：x = 2。

---

8. Sum and Product of Roots | 根的和与积

For a quadratic equation ax² + bx + c = 0 with roots α and β, there are elegant relationships between the roots and the coefficients:

对于根为 α 和 β 的二次方程 ax² + bx + c = 0，根与系数之间存在优美的关系：

Sum of roots: α + β = −b ÷ a

Product of roots: α × β = c ÷ a

These relationships allow you to form a quadratic equation from given roots. If the roots are α and β, the equation is:

利用这些关系，可以从已知的根构造二次方程。若根为 α 和 β，则方程为：

x² − (α + β)x + αβ = 0

For instance, if the roots are 3 and −2, then the sum is 1 and the product is −6, giving x² − x − 6 = 0.

例如，若根为 3 和 −2，则和为 1，积为 −6，得到方程 x² − x − 6 = 0。

---

9. Sketching Quadratic Graphs | 绘制二次函数图像

To sketch the graph of a quadratic function accurately, you need to identify four key features:

要准确绘制二次函数的图像，你需要确定四个关键特征：

• The y-intercept: set x = 0, giving y = c. | y 轴截距：令 x = 0，得 y = c。
• The x-intercepts: solve ax² + bx + c = 0 (if real roots exist). | x 轴截距：解 ax² + bx + c = 0（若存在实数根）。
• The vertex: found by completing the square or using x = −b ÷ (2a). | 顶点：通过配方或 x = −b ÷ (2a) 求得。
• The direction of opening: determined by the sign of a. | 开口方向：由 a 的符号决定。

Let us sketch y = x² − 2x − 3. The y-intercept is −3. Factorising gives (x − 3)(x + 1), so the x-intercepts are 3 and −1. The vertex has x = 1, giving y = −4. With a > 0, the parabola opens upward.

我们绘制 y = x² − 2x − 3。y 轴截距为 −3。因式分解得 (x − 3)(x + 1)，因此 x 轴截距为 3 和 −1。顶点的 x = 1，代入得 y = −4。由于 a > 0，抛物线开口向上。

---

10. Quadratic Inequalities | 二次不等式

Solving quadratic inequalities requires understanding the sign of the quadratic expression in different intervals. Consider x² − x − 6 > 0.

求解二次不等式需要理解二次表达式在不同区间内的正负号。考虑 x² − x − 6 > 0。

First factorise: (x − 3)(x + 2) > 0. The critical points are x = 3 and x = −2. Test a value in each interval:

首先因式分解：(x − 3)(x + 2) > 0。临界点为 x = 3 和 x = −2。在每个区间内取一个测试值：

• For x < −2, say x = −3: (−6)(−1) = 6 > 0 ✓ | 当 x < −2，取 x = −3：(−6)(−1) = 6 > 0 ✓
• For −2 < x < 3, say x = 0: (−3)(2) = −6 < 0 ✗ | 当 −2 < x < 3，取 x = 0：(−3)(2) = −6 < 0 ✗
• For x > 3, say x = 4: (1)(6) = 6 > 0 ✓ | 当 x > 3，取 x = 4：(1)(6) = 6 > 0 ✓

Therefore, the solution is x < −2 or x > 3.

因此，解集为 x < −2 或 x > 3。

---

11. Applications in Problem Solving | 实际应用与解题

Quadratic equations appear frequently in problem-solving contexts. A classic example involves projectile motion: an object is launched upward with its height modelled by h(t) = −5t² + 20t + 1, where h is in metres and t is in seconds.

二次方程经常出现在实际应用中。一个经典例子涉及抛体运动：一个物体被向上抛出，其高度由 h(t) = −5t² + 20t + 1 建模，其中 h 以米为单位，t 以秒为单位。

To find when the object hits the ground, set h(t) = 0:

要求物体何时落地，令 h(t) = 0：

−5t² + 20t + 1 = 0

Using the quadratic formula:

使用求根公式：

t = (−20 ± √(400 + 20)) ÷ (−10) = (−20 ± √420) ÷ (−10)

The positive root gives t ≈ 4.05 s. The negative root is discarded as time cannot be negative.

正根给出 t ≈ 4.05 秒。负根因时间不能为负数而被舍去。

Other applications include area problems, number puzzles, and cost optimisation.

其他应用包括面积问题、数字谜题以及成本优化问题。

---

12. Common Mistakes and Examination Tips | 常见错误与备考建议

Students frequently lose marks in quadratic questions due to avoidable errors. Being aware of these pitfalls will help you perform better in your IGCSE examination.

学生在二次方程题目中经常因可避免的错误而失分。了解这些陷阱将帮助你在 IGCSE 考试中发挥更佳。

• Forgetting to rearrange the equation to standard form before solving. | 在求解前忘记将方程整理为标准形式。
• Losing a root when dividing both sides by a common factor containing x. | 两边同时除以含有 x 的公因式时丢失一个根。
• Misapplying the quadratic formula by confusing the signs of b. | 误用求根公式，搞错 b 的符号。
• Forgetting that a quadratic equation can have zero, one, or two real solutions. | 忘记二次方程可能有一个、两个或零个实数解。
• Sketching graphs without identifying the vertex and intercepts. | 画图时未标出顶点和截距。

Top tip: Always check your solutions by substituting them back into the original equation. In exams, show every step of your working — method marks are awarded even when the final answer is incorrect.

备考提示：始终将解代回原方程进行验算。在考试中，写出每一步过程——即使最终答案有误，方法分仍会被给予。

---

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）