Understanding Quadratic Equations and Their Graphs | 理解二次方程及其图像

📚 Understanding Quadratic Equations and Their Graphs | 理解二次方程及其图像

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, coordinate geometry, and even problem-solving questions. Mastering quadratics not only boosts your exam score but also builds a strong foundation for A-Level Mathematics.

二次方程是 IGCSE 数学中最重要的主题之一。它出现在代数、图像、坐标几何甚至应用题中。掌握二次方程不仅能提高你的考试成绩，还能为 A-Level 数学打下坚实基础。

1. What Is a Quadratic Equation? | 什么是二次方程？

A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. Its general form is written as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

二次方程是次数为 2 的多项式方程，意味着变量的最高次数是 2。它的一般形式写作 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。

For example, 2x² + 3x − 5 = 0 is a quadratic equation because the highest power of x is 2. The coefficient a must not be zero; if a = 0, the equation becomes linear, not quadratic.

例如，2x² + 3x − 5 = 0 是一个二次方程，因为 x 的最高次数是 2。系数 a 不能为零；如果 a = 0，方程就变成了一次方程，而不是二次方程。

Standard form: ax² + bx + c = 0
a is the coefficient of x² (leading coefficient)
b is the coefficient of x
c is the constant term

标准形式：ax² + bx + c = 0
a 是 x² 的系数（首项系数）
b 是 x 的系数
c 是常数项

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

Factorisation is often the fastest method for solving quadratic equations when the equation can be written as a product of two linear factors. The idea is to rewrite ax² + bx + c = 0 as (px + q)(rx + s) = 0. Then, using the zero product property, each factor can be set to zero.

当二次方程可以写成两个一次因式的乘积时，因式分解通常是最快的解法。思路是将 ax² + bx + c = 0 改写为 (px + q)(rx + s) = 0。然后利用零积性质，令每个因式等于零。

Example: Solve x² − 5x + 6 = 0.

示例：解方程 x² − 5x + 6 = 0。

We look for two numbers that multiply to 6 and add to −5. These numbers are −2 and −3. Therefore:

我们寻找两个数，它们相乘得 6，相加得 −5。这两个数是 −2 和 −3。因此：

x² − 5x + 6 = (x − 2)(x − 3) = 0

So x − 2 = 0 or x − 3 = 0, giving x = 2 or x = 3.

所以 x − 2 = 0 或 x − 3 = 0，得到 x = 2 或 x = 3。

For equations where the leading coefficient a is not 1, factorisation requires a bit more care. For example, 2x² + 5x + 3 = 0 can be factorised as (2x + 3)(x + 1) = 0.

当首项系数 a 不等于 1 时，因式分解需要更加小心。例如，2x² + 5x + 3 = 0 可以因式分解为 (2x + 3)(x + 1) = 0。

3. Solving by Completing the Square | 用配方法解二次方程

Completing the square is a powerful method that works for any quadratic equation, even when factorisation is difficult or impossible. The method rewrites ax² + bx + c as a(x + p)² + q, which makes solving straightforward.

配方法是一种强大的解法，适用于任何二次方程，即使因式分解困难或不可行。该方法将 ax² + bx + c 改写为 a(x + p)² + q，从而使得求解变得直接。

Example: Solve x² + 6x + 2 = 0 by completing the square.

示例：用配方法解方程 x² + 6x + 2 = 0。

First, take half of 6, which is 3, and write:

首先，取 6 的一半，即 3，然后写出：

x² + 6x = (x + 3)² − 9

Therefore the equation becomes (x + 3)² − 9 + 2 = 0, so (x + 3)² = 7. Taking square roots gives x + 3 = ±√7, hence x = −3 ± √7.

因此方程变为 (x + 3)² − 9 + 2 = 0，所以 (x + 3)² = 7。两边开平方得 x + 3 = ±√7，因此 x = −3 ± √7。

If the coefficient a is not 1, factor a out first, then complete the square inside the bracket.

如果系数 a 不等于 1，先提出 a，再在括号内配方。

4. The Quadratic Formula | 二次公式

The quadratic formula is a universal tool that solves any quadratic equation of the form ax² + bx + c = 0. It is especially useful when factorisation is not straightforward. The formula is:

二次公式是解任意形式 ax² + bx + c = 0 的通用工具。当因式分解不直接时，它尤其有用。公式如下：

x = (−b ± √(b² − 4ac)) / 2a

Example: Solve 2x² − 4x − 3 = 0 using the quadratic formula.

示例：用二次公式解 2x² − 4x − 3 = 0。

Here a = 2, b = −4, c = −3. Substituting into the formula:

这里 a = 2，b = −4，c = −3。代入公式：

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = (4 ± 2√10) / 4

So x = 1 ± (√10)/2.

所以 x = 1 ± (√10)/2。

Notice that the expression b² − 4ac is called the discriminant. It determines the nature of the roots.

注意 b² − 4ac 被称为判别式。它决定了根的性质。

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

The discriminant, usually denoted by Δ, is defined as Δ = b² − 4ac. It tells us how many real roots a quadratic equation has, without actually solving the equation.

判别式通常用 Δ 表示，定义为 Δ = b² − 4ac。它告诉我们二次方程有多少个实根，而无需真正解方程。

Discriminant Δ	Nature of roots	Graph interpretation
Δ > 0	Two distinct real roots	Graph crosses the x-axis twice
Δ = 0	One repeated real root	Graph touches the x-axis once
Δ < 0	No real roots	Graph does not touch the x-axis

判别式 Δ	根的性质	图像含义
Δ > 0	两个不同的实根	图像与 x 轴相交两次
Δ = 0	一个重根	图像与 x 轴相切一次
Δ < 0	没有实根	图像不与 x 轴相交

For example, the equation x² + 2x + 5 = 0 has Δ = 4 − 20 = −16, which is negative, so it has no real roots.

例如，方程 x² + 2x + 5 = 0 的判别式 Δ = 4 − 20 = −16，是负数，因此它没有实根。

6. 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 parabola. If a > 0, the parabola opens upwards and has a minimum point. If a < 0, it opens downwards and has a maximum point.

二次函数 y = ax² + bx + c 的图像是一条抛物线。如果 a > 0，抛物线开口向上，有最小值点；如果 a < 0，抛物线开口向下，有最大值点。

The vertex, also called the turning point, is the key feature of a parabola. Its x-coordinate is given by x = −b/(2a). The y-coordinate can be found by substituting this x-value into the equation.

顶点，也称为转向点，是抛物线的重要特征。它的 x 坐标由 x = −b/(2a) 给出。将 x 值代入方程即可求出 y 坐标。

For example, the parabola y = x² − 4x + 3 has vertex at x = 4/2 = 2, and y = 2² − 8 + 3 = −1. So the vertex is (2, −1).

例如，抛物线 y = x² − 4x + 3 的顶点在 x = 4/2 = 2 处，y = 2² − 8 + 3 = −1。所以顶点是 (2, −1)。

7. Sketching Quadratic Graphs | 画二次函数草图

Sketching a quadratic graph requires identifying four key features: the y-intercept, the x-intercepts (if any), the vertex, and the direction of the parabola. Once these are known, the graph can be drawn accurately enough for most exam questions.

画二次函数草图需要确定四个关键特征：y 截距、x 截距（如果有）、顶点以及抛物线开口方向。一旦知道这些，就可以画出足以应对大多数考试题目的图像。

For y = x² − 4x + 3:

对于 y = x² − 4x + 3：

y-intercept: set x = 0, so y = 3; the point is (0, 3)
x-intercepts: solve x² − 4x + 3 = 0, so (x − 1)(x − 3) = 0, giving x = 1 and x = 3
Vertex: x = 2, y = −1
Opens upwards because a = 1 > 0

y 截距：令 x = 0，得 y = 3；点为 (0, 3)
x 截距：解 x² − 4x + 3 = 0，得 (x − 1)(x − 3) = 0，所以 x = 1 和 x = 3
顶点：x = 2，y = −1
开口向上，因为 a = 1 > 0

With these four features, you can draw a smooth curve passing through the points, forming a U-shaped parabola.

有了这四个特征，你可以画一条平滑曲线穿过这些点，形成 U 形抛物线。

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 useful relationships between the roots and the coefficients:

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

α + β = −b/a

αβ = c/a

α + β = −b/a

αβ = c/a

These relationships allow you to find the sum and product of roots without solving the equation. For example, for 2x² + 3x − 5 = 0, the sum of roots is −3/2 and the product is −5/2.

这些关系允许你不解方程就能求出根的和与积。例如，对于 2x² + 3x − 5 = 0，根的和是 −3/2，根的积是 −5/2。

You can also construct a quadratic equation from given roots. If the roots are α and β, the equation is x² − (α + β)x + αβ = 0.

你还可以根据给定的根构造二次方程。如果根是 α 和 β，方程为 x² − (α + β)x + αβ = 0。

9. Solving Quadratic Inequalities | 解二次不等式

Quadratic inequalities involve expressions like ax² + bx + c > 0 or ax² + bx + c ≤ 0. To solve them, first solve the corresponding quadratic equation to find critical points, then test intervals on a number line.

二次不等式涉及形如 ax² + bx + c > 0 或 ax² + bx + c ≤ 0 的表达式。解这类不等式，首先解对应的二次方程以找到临界点，然后在数轴上测试区间。

Example: Solve x² − x − 6 > 0.

示例：解不等式 x² − x − 6 > 0。

First solve x² − x − 6 = 0, which gives (x − 3)(x + 2) = 0, so x = 3 or x = −2. These two values divide the number line into three intervals: x < −2, −2 < x < 3, and x > 3.

首先解 x² − x − 6 = 0，得到 (x − 3)(x + 2) = 0，所以 x = 3 或 x = −2。这两个值将数轴分为三个区间：x < −2、−2 < x < 3 和 x > 3。

Testing x = −3 gives (−3)² − (−3) − 6 = 6 > 0, so the inequality holds for x < −2. Testing x = 0 gives −6 < 0, so the middle interval does not work. Testing x = 4 gives 16 − 4 − 6 = 6 > 0, so x > 3 works. Therefore the solution is x < −2 or x > 3.

测试 x = −3 得到 (−3)² − (−3) − 6 = 6 > 0，所以不等式在 x < −2 时成立。测试 x = 0 得 −6 < 0，所以中间区间不成立。测试 x = 4 得 16 − 4 − 6 = 6 > 0，所以 x > 3 成立。因此解为 x < −2 或 x > 3。

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

Many real-world problems can be modelled by quadratic equations. Common examples include area problems, projectile motion, and number puzzles. The key is to translate the given information into a quadratic equation and then solve it.

许多现实问题可以用二次方程建模。常见例子包括面积问题、抛体运动和数字谜题。关键是将给定信息转化为二次方程并求解。

Example: A rectangle has a length that is 3 cm longer than its width. Its area is 40 cm². Find the width.

示例：一个矩形的长比宽长 3 cm，面积为 40 cm²。求宽。

Let the width be x cm. Then the length is x + 3 cm. The area is x(x + 3) = 40, which gives x² + 3x − 40 = 0. Factorising: (x + 8)(x − 5) = 0, so x = −8 or x = 5. Since width cannot be negative, the width is 5 cm.

设宽为 x cm，则长为 x + 3 cm。面积为 x(x + 3) = 40，即 x² + 3x − 40 = 0。因式分解：(x + 8)(x − 5) = 0，所以 x = −8 或 x = 5。由于宽不能为负，所以宽为 5 cm。

Always check whether your solutions make sense in the context of the problem. Negative lengths, times, or quantities should be rejected.

始终检查你的解在问题的实际背景中是否有意义。负的长度、时间或数量应当舍去。

11. Common Mistakes and Tips | 常见错误与提示

Students often make predictable mistakes when working with quadratics. Recognising these errors can save valuable marks in exams.

学生在处理二次方程时经常犯一些可预测的错误。识别这些错误可以在考试中节省宝贵的分数。

Forgetting to set the equation to zero before factorising or using the formula
Misplacing signs when substituting into the quadratic formula
Forgetting that a negative discriminant means no real roots, not an error
Confusing the x-coordinate of the vertex with the roots
Neglecting to reject invalid negative answers in word problems

在因式分解或使用公式之前忘记将方程化为零
代入二次公式时弄错符号
忘记判别式为负意味着没有实根，而不是计算出错
将顶点的 x 坐标与根混淆
在应用题中忘记舍去无效的负数答案

To avoid these, always write the equation in standard form first, double-check your signs, and read word problems carefully to identify the correct variable and conditions.

为避免这些错误，始终先将方程写成标准形式，仔细检查符号，并认真阅读应用题以确定正确的变量和条件。

12. Practice Questions | 练习题目

Try the following questions to test your understanding. Full solutions are not provided here, but you can check your answers by substituting them back into the original equations.

尝试以下题目以检验你的理解。这里不提供完整解答，但你可以通过代入原方程来检查答案。

1. Solve x² − 7x + 10 = 0 by factorisation.

1. 用因式分解解 x² − 7x + 10 = 0。

2. Solve 2x² + 3x − 2 = 0 using the quadratic formula.

2. 用二次公式解 2x² + 3x − 2 = 0。

3. Find the vertex of y = −x² + 4x + 1 and state whether it is a maximum or minimum.

3. 求 y = −x² + 4x + 1 的顶点，并说明它是最大值还是最小值。

4. Determine the value of k for which x² + kx + 9 = 0 has exactly one real root.

4. 求使 x² + kx + 9 = 0 恰好有一个实根时的 k 值。

5. Solve the inequality x² − 4x − 5 ≤ 0.

5. 解不等式 x² − 4x − 5 ≤ 0。

6. A rectangle has perimeter 20 cm and area 21 cm². Find its dimensions.

6. 一个矩形的周长为 20 cm，面积为 21 cm²。求它的尺寸。

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Understanding Quadratic Equations and Their Graphs | 理解二次方程及其图像