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

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

Quadratic functions and equations form a cornerstone of the IGCSE Mathematics syllabus. From factorisation to the quadratic formula, mastering these techniques is essential for solving real-world problems and progressing to advanced topics.

二次函数与二次方程是 IGCSE 数学教学大纲中的基石内容。从因式分解到求根公式，掌握这些技巧对于解决实际问题以及学习更高级的课题至关重要。

1. What Is a Quadratic Expression? | 什么是二次表达式？

A quadratic expression in one variable x is written in the general form ax² + bx + c, where a, b and c are constants and a ≠ 0. The highest power of x is 2, which gives the expression its name.

含有一个变量 x 的二次表达式写作一般形式 ax² + bx + c，其中 a、b、c 是常数且 a ≠ 0。x 的最高次数是 2，这也是“二次”名称的由来。

Examples of quadratic expressions include x² − 5x + 6, 2x² + 3x − 1 and 4x² − 9. Notice that the coefficient a can be any non-zero number.

二次表达式的例子包括 x² − 5x + 6、2x² + 3x − 1 和 4x² − 9。请注意，系数 a 可以是任何非零数字。

If a = 0, the expression becomes linear, not quadratic: bx + c.

如果 a = 0，表达式变为一次式而非二次式：bx + c。
The constant term c may be zero, for example x² + 2x = x(x + 2).

常数项 c 可以为零，例如 x² + 2x = x(x + 2)。

2. Standard Form and Key Features | 标准形式与关键特征

The standard form of a quadratic equation is ax² + bx + c = 0. This form is important because it allows us to apply the quadratic formula or the discriminant directly.

二次方程的标准形式是 ax² + bx + c = 0。这种形式非常重要，因为我们可以直接应用求根公式或判别式。

In the graph of y = ax² + bx + c, the sign of a determines the direction of the curve: if a > 0, the parabola opens upward (a U-shape); if a < 0, it opens downward (an n-shape).

在 y = ax² + bx + c 的图像中，a 的符号决定曲线的开口方向：如果 a > 0，抛物线向上开口（U 形）；如果 a < 0，则向下开口（n 形）。

y = ax² + bx + c, with a ≠ 0

y = ax² + bx + c，其中 a ≠ 0

The y-intercept is always the point (0, c), since substituting x = 0 gives y = c.

y 轴截距始终是点 (0, c)，因为令 x = 0 时 y = c。

3. Solving by Factorisation | 因式分解法

When a quadratic equation can be factorised, this method is often the fastest. We look for two numbers whose product is ac and whose sum is b when a ≠ 1, or simpler pairs when a = 1.

当二次方程可以因式分解时，这种方法通常最快。当 a ≠ 1 时，我们寻找两个数，使它们的乘积为 ac、和为 b；当 a = 1 时则寻找更简单的一对数。

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

以 x² − 5x + 6 = 0 为例，我们需要找到两个数，它们相乘为 6、相加为 −5，这就是 −2 和 −3。

x² − 5x + 6 = (x − 2)(x − 3) = 0

By the zero product property, if (x − 2)(x − 3) = 0, then x = 2 or x = 3. Therefore the solutions are x = 2 and x = 3.

根据零乘积性质，若 (x − 2)(x − 3) = 0，则 x = 2 或 x = 3。因此方程的解为 x = 2 和 x = 3。

Always rearrange the equation to the form ax² + bx + c = 0 before factorising.

因式分解前务必先将方程整理为 ax² + bx + c = 0 的形式。
If the product of two factors is zero, at least one factor must be zero.

如果两个因式的乘积为零，那么至少有一个因式为零。

4. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This form is useful for finding the vertex of a parabola and for solving equations when factorisation is difficult.

配方法将二次式改写为 a(x + p)² + q 的形式。这种形式有助于找到抛物线的顶点，也可在因式分解困难时解方程。

Start with x² + 6x + 5 = 0. Take half of 6, which is 3, and square it to get 9. Then add and subtract 9 inside the expression.

以 x² + 6x + 5 = 0 为例，取 6 的一半为 3，平方得到 9，然后在表达式中加上并减去 9。

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

So x² + 6x + 5 = 0 becomes (x + 3)² − 4 = 0, giving (x + 3)² = 4. Taking square roots gives x + 3 = ±2, so x = −1 or x = −5.

因此 x² + 6x + 5 = 0 变为 (x + 3)² − 4 = 0，即 (x + 3)² = 4。两边开平方得 x + 3 = ±2，所以 x = −1 或 x = −5。

When the coefficient a is not 1, factor it out first: 2x² + 8x + 3 = 2(x² + 4x) + 3, then complete the square inside the bracket.

当 a 不等于 1 时，先提出 a：2x² + 8x + 3 = 2(x² + 4x) + 3，再在括号内配方。

5. The Quadratic Formula | 求根公式

The quadratic formula provides a universal method for solving any quadratic equation. For ax² + bx + c = 0, the solutions are given by this expression.

求根公式（二次公式）为解任何二次方程提供了一种通用方法。对于 ax² + bx + c = 0，解由以下表达式给出。

x = [−b ± √(b² − 4ac)] / 2a

For example, solve 2x² − 4x − 3 = 0. Here a = 2, b = −4 and c = −3. Substituting these values into the formula gives the following working.

例如，解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。将这些数值代入公式后得到以下运算过程。

x = [4 ± √((−4)² − 4 × 2 × (−3))] / (2 × 2) = [4 ± √(16 + 24)] / 4 = [4 ± √40] / 4

Since √40 = 2√10, the exact solutions are x = (2 ± √10) / 2. Approximation to two decimal places gives x ≈ 2.58 or x ≈ −0.58.

因为 √40 = 2√10，精确解为 x = (2 ± √10) / 2。保留两位小数得 x ≈ 2.58 或 x ≈ −0.58。

6. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: b² − 4ac. Its value tells us how many real roots the equation has.

判别式是求根公式中根号下的部分：b² − 4ac。它的值告诉我们方程有多少个实数根。

Value of b² − 4ac	Number of real roots	Graph interpretation
Positive (b² − 4ac > 0)	Two distinct real roots	Parabola crosses the x-axis at two points
Zero (b² − 4ac = 0)	One repeated real root	Parabola touches the x-axis at one point
Negative (b² − 4ac < 0)	No real roots	Parabola does not cross the x-axis
b² − 4ac 的值	实数根的个数	图像特征
正数（b² − 4ac > 0）	两个不同的实数根	抛物线与 x 轴有两个交点
零（b² − 4ac = 0）	一个重根	抛物线与 x 轴相切于一点
负数（b² − 4ac < 0）	无实数根	抛物线与 x 轴不相交

For the equation x² + 2x + 5 = 0, the discriminant is 2² − 4 × 1 × 5 = 4 − 20 = −16, which is negative. Therefore the equation has no real solutions.

对于方程 x² + 2x + 5 = 0，判别式为 2² − 4 × 1 × 5 = 4 − 20 = −16，是负数，因此该方程没有实数解。

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

The graph of y = ax² + bx + c is a smooth curve called a parabola. To sketch it accurately, you need the line of symmetry, the vertex and the x-intercepts.

y = ax² + bx + c 的图像是一条平滑曲线，称为抛物线。要准确作图，你需要对称轴、顶点和 x 轴截距。

The line of symmetry is x = −b / 2a.

对称轴是 x = −b / 2a。
The vertex has x-coordinate −b / 2a. Substitute this back into the equation to find the y-coordinate.

顶点的 x 坐标为 −b / 2a，将其代回方程即可求得 y 坐标。
The x-intercepts (if they exist) are the roots of ax² + bx + c = 0.

x 轴截距（如果存在）就是 ax² + bx + c = 0 的根。

Consider y = x² − 4x + 3. The line of symmetry is x = 2, and substituting gives y = 4 − 8 + 3 = −1, so the vertex is (2, −1). Factoring gives (x − 1)(x − 3), so x-intercepts are 1 and 3.

以 y = x² − 4x + 3 为例，对称轴为 x = 2，代入得 y = 4 − 8 + 3 = −1，因此顶点为 (2, −1)。因式分解得 (x − 1)(x − 3)，所以 x 截距为 1 和 3。

8. Finding Roots from the Graph | 从图像中求根

Given a graph of y = ax² + bx + c, the roots of the equation ax² + bx + c = 0 are the x-coordinates where the graph crosses the x-axis.

如果给定 y = ax² + bx + c 的图像，那么方程 ax² + bx + c = 0 的根就是图像与 x 轴交点的 x 坐标。

If the graph does not cross the x-axis, the equation has no real roots, and the discriminant is negative.

如果图像不与 x 轴相交，方程就没有实数根，判别式为负。

To solve a related equation such as x² − 4x + 3 = 1, draw the line y = 1 on the same graph and read the x-coordinates of the intersection points.

要解相关方程如 x² − 4x + 3 = 1，可在同一坐标系中画出直线 y = 1，然后读取交点处的 x 坐标。

Roots ↔ x-intercepts ↔ factors ↔ zeros

根 ↔ x 轴截距 ↔ 因式 ↔ 零点

These four terms describe the same idea from different perspectives: values of x that make the quadratic equal to zero.

这四个术语从不同角度描述同一概念：使二次式等于零的 x 值。

9. Applications in Problem Solving | 实际应用问题

Quadratics appear in geometry, physics and economics. A rectangle with length x and width x − 3 has area 28, giving the equation x(x − 3) = 28.

二次函数出现在几何、物理和经济等领域。一个长为 x、宽为 x − 3 的矩形面积为 28，由此得到方程 x(x − 3) = 28。

x² − 3x − 28 = 0 ⇒ (x − 7)(x + 4) = 0

The solutions are x = 7 and x = −4. Length cannot be negative, so x = 7 and width = 4. Always reject solutions that do not make sense in context.

解为 x = 7 和 x = −4。长度不能为负数，所以 x = 7，宽为 4。一定要注意舍去在情境中没有意义的解。

Read the question carefully to identify which variable to define.

仔细阅读题目，确定要定义的变量。
Form the quadratic equation from the given conditions.

根据已知条件列出二次方程。
Check whether the answer satisfies the original word problem.

检查答案是否满足原始文字题的意义。

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

Students often forget to rearrange the equation before factoring, or they lose the negative sign when substituting b into the quadratic formula.

学生常犯的错误包括：在因式分解前忘记整理方程，或将 b 代入求根公式时漏掉负号。

Always write the equation in standard form first: ax² + bx + c = 0.

始终先将方程写成标准形式：ax² + bx + c = 0。
When b is negative, use a bracket in the formula, such as x = [−(−4) ± √(−4)² − 4ac] / 2a.

当 b 为负数时，在公式中使用括号，例如 x = [−(−4) ± √(−4)² − 4ac] / 2a。
Show your substitution clearly; IGCSE mark schemes award method marks even for incorrect arithmetic.

清楚地展示代入过程；IGCSE 评分标准会对方法步骤给分，即使算术出错也可能获得方法分。
If a question asks for exact roots, leave √ in your answer; do not round unless instructed.

如果题目要求精确根，请在答案中保留根号；除非题目要求，否则不要四舍五入。
Use the discriminant to check the number of solutions before solving.

在求解之前先用判别式判断解的个数。

Practice each method separately, then choose the most efficient one for each question. Factorisation is quickest, the quadratic formula always works, and completing the square is essential for vertex problems.

分别练习每种方法，然后在具体题目中选择最有效的方法。因式分解最快，求根公式通用性最强，而配方法对于顶点问题必不可少。

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