Solving Quadratic Equations and Functions | 解二次方程与二次函数

📚 Solving Quadratic Equations and Functions | 解二次方程与二次函数

Quadratic equations and functions are a core part of the IGCSE Edexcel Mathematics syllabus. This article explains the essential techniques, from factorising to using the quadratic formula, and shows how to interpret graphs of quadratic functions.

二次方程与二次函数是 IGCSE Edexcel 数学考纲中的核心内容。本文将讲解从因式分解到使用求根公式的关键方法，并说明如何理解二次函数的图像。

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

A quadratic expression in the variable \(x\) is written in the form \(ax^2 + bx + c\), where \(a, b, c\) are constants and \(a \neq 0\). The highest power of \(x\) is 2.

含变量 \(x\) 的二次表达式写作 \(ax^2 + bx + c\)，其中 \(a, b, c\) 为常数，且 \(a \neq 0\)。\(x\) 的最高次数为 2。

The coefficient of \(x^2\) is called \(a\).

\(x^2\) 的系数称为 \(a\)。
The constant term is \(c\).

常数项是 \(c\)。
If \(a = 0\), the expression becomes linear, not quadratic.

如果 \(a = 0\)，表达式变为一次式，而不是二次式。

ax² + bx + c = 0

Examples of quadratic equations: \(x^2 – 5x + 6 = 0\), \(2x^2 + 3x – 2 = 0\), and \(x^2 = 9\).

二次方程的例子：\(x^2 – 5x + 6 = 0\)、\(2x^2 + 3x – 2 = 0\) 和 \(x^2 = 9\)。

2. Expanding and Factorising Quadratics | 展开与因式分解二次式

Expanding means removing brackets, while factorising means writing as a product of two linear factors. These are inverse processes.

展开是指去括号，因式分解是写成两个一次因式的乘积。两者互为逆运算。

For example, expanding \((x + 3)(x – 2)\) gives \(x^2 + x – 6\). Therefore, factorising \(x^2 + x – 6\) gives \((x + 3)(x – 2)\).

例如，展开 \((x + 3)(x – 2)\) 得到 \(x^2 + x – 6\)。因此，因式分解 \(x^2 + x – 6\) 得到 \((x + 3)(x – 2)\)。

(x + p)(x + q) = x² + (p + q)x + pq

When factorising \(x^2 + bx + c\), find two numbers whose product is \(c\) and whose sum is \(b\).

因式分解 \(x^2 + bx + c\) 时，找到两个数，使它们的乘积为 \(c\)，和为 \(b\)。

Expression	Factorised form
x² + 5x + 6	(x + 2)(x + 3)
x² − 4	(x − 2)(x + 2)
2x² + 7x + 3	(2x + 1)(x + 3)

3. Solving by Factorisation | 用因式分解法解方程

If a quadratic equation can be factorised, we use the zero product property: if \(ab = 0\), then \(a = 0\) or \(b = 0\).

如果二次方程可以因式分解，我们使用零积性质：若 \(ab = 0\)，则 \(a = 0\) 或 \(b = 0\)。

Example: solve \(x^2 – 5x + 6 = 0\).

例：解 \(x^2 – 5x + 6 = 0\)。

Factorise: \((x – 2)(x – 3) = 0\).

因式分解：\((x – 2)(x – 3) = 0\)。

So \(x – 2 = 0\) or \(x – 3 = 0\), giving \(x = 2\) or \(x = 3\).

因此 \(x – 2 = 0\) 或 \(x – 3 = 0\)，得 \(x = 2\) 或 \(x = 3\)。

Always rearrange the equation to the form \(ax^2 + bx + c = 0\) before factorising.

因式分解前，务必先把方程整理成 \(ax^2 + bx + c = 0\) 的形式。

4. Solving by Completing the Square | 用配方法解方程

Completing the square rewrites a quadratic as \(a(x + p)^2 + q\). This is useful when factorisation is difficult.

配方法将二次式改写为 \(a(x + p)^2 + q\)。当因式分解困难时，此法非常有用。

For \(x^2 + bx + c\), complete the square using the formula:

对于 \(x^2 + bx + c\)，配方公式为：

x² + bx + c = (x + b/2)² − (b/2)² + c

Example: solve \(x^2 + 6x + 1 = 0\).

例：解 \(x^2 + 6x + 1 = 0\)。

Complete the square: \((x + 3)^2 – 9 + 1 = 0\), so \((x + 3)^2 = 8\).

配方：\((x + 3)^2 – 9 + 1 = 0\)，所以 \((x + 3)^2 = 8\)。

Take square roots: \(x + 3 = ±√8\), hence \(x = -3 ± 2√2\).

开平方：\(x + 3 = ±√8\)，因此 \(x = -3 ± 2√2\)。

5. The Quadratic Formula | 二次求根公式

The quadratic formula solves any quadratic equation \(ax^2 + bx + c = 0\) directly.

二次求根公式可直接解任意形式 \(ax^2 + bx + c = 0\) 的二次方程。

x = (−b ± √(b² − 4ac)) / (2a)

Substitute the values of \(a\), \(b\), and \(c\) into the formula. Be careful with negative signs.

将 \(a\)、\(b\)、\(c\) 的值代入公式。注意负号的处理。

Example: solve \(2x^2 + 3x – 2 = 0\). Here \(a = 2\), \(b = 3\), \(c = -2\).

例：解 \(2x^2 + 3x – 2 = 0\)。这里 \(a = 2\)，\(b = 3\)，\(c = -2\)。

\(x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4\).

\(x = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4\)。

So \(x = 1/2\) or \(x = -2\).

所以 \(x = 1/2\) 或 \(x = -2\)。

6. The Discriminant | 判别式

The discriminant is the part \(b^2 – 4ac\). It tells us how many real roots a quadratic equation has.

判别式是 \(b^2 – 4ac\) 部分，它告诉我们二次方程有多少个实数根。

Discriminant	Number of real roots
b² − 4ac > 0	Two distinct real roots
b² − 4ac = 0	One repeated real root
b² − 4ac < 0	No real roots

The discriminant also helps in sketching graphs and in problems involving tangents.

判别式也有助于画函数图像，以及解决与切线相关的问题。

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

The graph of \(y = ax^2 + bx + c\) is a parabola. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.

\(y = ax^2 + bx + c\) 的图像是抛物线。若 \(a > 0\)，抛物线开口向上；若 \(a < 0\)，开口向下。

The y-intercept is the point \((0, c)\).

y 轴截距是点 \((0, c)\)。
The x-intercepts are the real roots of the equation.

x 轴截距是方程的实数根。
The vertex (turning point) is the maximum or minimum point.

顶点（转向点）是最大值或最小值点。

The axis of symmetry is the vertical line through the vertex.

对称轴是经过顶点的竖直直线。

8. Roots, Intercepts and Turning Points | 根、截距与转向点

For \(y = x^2 – 4x + 3\), factorise to find roots: \((x – 1)(x – 3) = 0\), so \(x = 1\) and \(x = 3\).

对于 \(y = x^2 – 4x + 3\)，因式分解求根：\((x – 1)(x – 3) = 0\)，所以 \(x = 1\) 和 \(x = 3\)。

The y-intercept is \(y = 3\). The turning point lies halfway between the roots, at \(x = 2\). Substituting gives \(y = 4 – 8 + 3 = -1\).

y 轴截距为 \(y = 3\)。转向点位于两根中点，即 \(x = 2\)。代入得 \(y = 4 – 8 + 3 = -1\)。

So the vertex is \((2, -1)\). Since \(a = 1 > 0\), this is a minimum point.

因此顶点是 \((2, -1)\)。由于 \(a = 1 > 0\)，这是最小值点。

9. Sketching Graphs | 画函数图像简图

To sketch \(y = ax^2 + bx + c\), follow these steps:

画 \(y = ax^2 + bx + c\) 的简图，可按以下步骤：

Determine the direction of the parabola from the sign of \(a\).

根据 \(a\) 的符号判断抛物线开口方向。
Write the y-intercept \((0, c)\).

标出 y 轴截距 \((0, c)\)。
Solve \(ax^2 + bx + c = 0\) to find x-intercepts, if they exist.

解 \(ax^2 + bx + c = 0\) 求 x 轴截距（如果存在）。
Find the turning point using \(x = -b/(2a)\).

用 \(x = -b/(2a)\) 求转向点。

Label all key points clearly on your sketch.

在简图上清楚地标出所有关键点。

10. Word Problems | 应用题

Quadratic equations often appear in area, projectile motion, and number problems.

二次方程常出现在面积、抛体运动和数字问题中。

Example: The length of a rectangle is 3 cm longer than its width. Its area is 40 cm². Find the width.

例：一个长方形的长比宽多 3 cm，面积为 40 cm²。求宽。

Let the width be \(x\). Then the length is \(x + 3\). So \(x(x + 3) = 40\).

设宽为 \(x\)，则长为 \(x + 3\)。因此 \(x(x + 3) = 40\)。

This gives \(x^2 + 3x – 40 = 0\), which factorises as \((x + 8)(x – 5) = 0\). Since \(x > 0\), \(x = 5\).

得到 \(x^2 + 3x – 40 = 0\)，因式分解为 \((x + 8)(x – 5) = 0\)。由于 \(x > 0\)，所以 \(x = 5\)。

11. Exam Tips | 考试提示

These common mistakes cost marks. Keep them in mind:

以下常见错误会丢分，请务必留意：

Always rearrange to \(ax^2 + bx + c = 0\) before solving.

解方程前务必整理成 \(ax^2 + bx + c = 0\) 的形式。
When using the quadratic formula, write down the values of \(a\), \(b\), \(c\) first.

使用求根公式时，先写下 \(a\)、\(b\)、\(c\) 的值。
Check answers by substituting them back into the original equation.

将答案代回原方程验算。
Do not forget that a quadratic can have no, one, or two solutions.

不要忘记二次方程可能无解、有一个解或两个解。

Practice all three solving methods — factorisation, completing the square, and the formula — so you can choose the fastest one.

练习三种解法——因式分解法、配方法、求根公式法——以便选择最快的方法。

12. Summary | 总结

Quadratic equations and functions are essential for IGCSE Edexcel Mathematics. You must be able to expand and factorise, solve using multiple methods, interpret the discriminant, sketch graphs, and apply these skills to word problems.

二次方程与二次函数是 IGCSE Edexcel 数学的重要组成部分。你必须掌握展开与因式分解、多种解法、判别式的意义、图像画法，以及将这些技能应用于应用题。

With consistent practice, these topics become straightforward and rewarding.

通过持续练习，这些知识点会变得简单而有趣。

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Solving Quadratic Equations and Functions | 解二次方程与二次函数