Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations are one of the most important topics in the Edexcel IGCSE Mathematics syllabus. They appear in many exam questions, from basic factorisation to more complex graph sketching and problem solving.

二次方程是 Edexcel IGCSE 数学课程中最重要的主题之一。它们出现在许多考试题目中，从基础因式分解到更复杂的图像绘制和问题求解。

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

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

二次方程是形如 \(ax^2 + bx + c = 0\) 的方程，其中 \(a\)、\(b\) 和 \(c\) 是常数，且 \(a \neq 0\)。变量的最高次数是 2。

For example, \(2x^2 – 3x + 1 = 0\) is quadratic, but \(x^3 – 2x = 0\) is not.

例如，\(2x^2 – 3x + 1 = 0\) 是二次方程，而 \(x^3 – 2x = 0\) 不是。

The general form \(ax^2 + bx + c = 0\) is essential for applying the quadratic formula and the discriminant.

一般形式 \(ax^2 + bx + c = 0\) 对于使用二次公式和判别式至关重要。

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

Before solving, you must master expanding and factorising. Expanding means removing brackets: \((x + 3)(x – 5) = x^2 – 2x – 15\).

在求解之前，你必须熟练掌握展开和因式分解。展开意味着去掉括号：\((x + 3)(x – 5) = x^2 – 2x – 15\)。

Factorising is the reverse process: find two brackets that multiply to give the quadratic.

因式分解是相反的过程：找到两个括号，使其相乘得到该二次式。

For \(x^2 + bx + c\), find two numbers whose product is \(c\) and sum is \(b\).
对于 \(x^2 + bx + c\)，找到两个数，其乘积为 \(c\)，和为 \(b\)。
For \(ax^2 + bx + c\) with \(a > 1\), use the method of splitting the middle term or the box method.
对于 \(a > 1\) 的 \(ax^2 + bx + c\)，使用拆中项法或表格法。

Example: Factorise \(x^2 – 7x + 12\). Two numbers: product 12, sum -7, so the numbers are -3 and -4.

示例：因式分解 \(x^2 – 7x + 12\)。两个数：乘积为 12，和为 -7，所以这两个数是 -3 和 -4。

x² − 7x + 12 = (x − 3)(x − 4)

3. Solving by Factorisation | 因式分解法求解

Once a quadratic is factorised, use the zero product property: if \(AB = 0\), then \(A = 0\) or \(B = 0\).

一旦二次式被因式分解，使用零乘积性质：若 \(AB = 0\)，则 \(A = 0\) 或 \(B = 0\)。

Example: Solve \(x^2 – 7x + 12 = 0\).

示例：解 \(x^2 – 7x + 12 = 0\)。

Factorised form: \((x – 3)(x – 4) = 0\).

因式分解形式：\((x – 3)(x – 4) = 0\)。

So \(x – 3 = 0\) gives \(x = 3\), and \(x – 4 = 0\) gives \(x = 4\).

因此 \(x – 3 = 0\) 得 \(x = 3\)，\(x – 4 = 0\) 得 \(x = 4\)。

Always check your solutions by substituting them back into the original equation.

始终通过将解代入原方程来检查你的答案。

4. Solving by Quadratic Formula | 公式法求解

Not all quadratics factorise easily. You can always use the quadratic formula.

并非所有二次式都能轻易因式分解。你始终可以使用二次公式。

For \(ax^2 + bx + c = 0\),

对于 \(ax^2 + bx + c = 0\)，

x = (−b ± √(b² − 4ac)) / (2a)

The formula is given in the Edexcel IGCSE formula sheet, but you must know how to use it correctly.

该公式在 Edexcel IGCSE 公式表中给出，但你必须知道如何正确使用它。

Example: Solve \(2x^2 + 3x – 5 = 0\). Here \(a = 2\), \(b = 3\), \(c = -5\).

示例：解 \(2x^2 + 3x – 5 = 0\)。这里 \(a = 2\)，\(b = 3\)，\(c = -5\)。

Substitute into the formula:

代入公式：

x = (−3 ± √(3² − 4×2×(−5))) / (2×2) = (−3 ± √(9 + 40)) / 4 = (−3 ± √49) / 4

So \(x = (−3 + 7)/4 = 1\) or \(x = (−3 − 7)/4 = −5/2\).

因此 \(x = (−3 + 7)/4 = 1\) 或 \(x = (−3 − 7)/4 = −5/2\)。

5. Solving by Completing the Square | 配方法求解

Completing the square rewrites \(ax^2 + bx + c\) in the form \(a(x + p)^2 + q\). This is very useful for finding turning points.

配方法将 \(ax^2 + bx + c\) 改写成 \(a(x + p)^2 + q\) 的形式。这对于求顶点非常有用。

For \(x^2 + bx\), add and subtract \((b/2)^2\).

对于 \(x^2 + bx\)，加上并减去 \((b/2)^2\)。

Example: Solve \(x^2 + 6x – 1 = 0\) by completing the square.

示例：用配方法解 \(x^2 + 6x – 1 = 0\)。

First, complete the square for \(x^2 + 6x\): the value added is \((6/2)^2 = 9\).

首先，对 \(x^2 + 6x\) 配方：加上的值为 \((6/2)^2 = 9\)。

x² + 6x − 1 = (x + 3)² − 9 − 1 = (x + 3)² − 10

Set equal to 0: \((x + 3)^2 – 10 = 0\).

令其等于 0：\((x + 3)^2 – 10 = 0\)。

(x + 3)² = 10

Take square roots: \(x + 3 = ±√10\), so \(x = -3 ± √10\).

两边开平方：\(x + 3 = ±√10\)，所以 \(x = -3 ± √10\)。

6. The Discriminant | 判别式

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

\(b^2 – 4ac\) 这部分称为判别式。它告诉我们二次方程有多少个实根。

If \(b^2 – 4ac > 0\), there are two distinct real roots.
如果 \(b^2 – 4ac > 0\)，方程有两个不同的实根。
If \(b^2 – 4ac = 0\), there is one repeated real root.
如果 \(b^2 – 4ac = 0\)，方程有一个重根。
If \(b^2 – 4ac < 0\), there are no real roots.
如果 \(b^2 – 4ac < 0\)，方程没有实根。

Example: How many real roots does \(3x^2 – 2x + 5 = 0\) have?

示例：\(3x^2 – 2x + 5 = 0\) 有多少个实根？

Discriminant: \((-2)^2 – 4×3×5 = 4 – 60 = -56 < 0\).

判别式：\((-2)^2 – 4×3×5 = 4 – 60 = -56 < 0\)。

So there are no real roots.

所以没有实根。

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

A quadratic function \(y = ax^2 + bx + c\) produces a parabola. The sign of \(a\) determines the shape:

二次函数 \(y = ax^2 + bx + c\) 的图像是抛物线。\(a\) 的符号决定其形状：

If \(a > 0\), the parabola opens upwards (U-shape).
如果 \(a > 0\)，抛物线开口向上（U 形）。
If \(a < 0\), the parabola opens downwards (∩-shape).
如果 \(a < 0\)，抛物线开口向下（∩ 形）。

The roots of the equation \(ax^2 + bx + c = 0\) are the x-intercepts of the graph.

方程 \(ax^2 + bx + c = 0\) 的根是图像与 x 轴的交点。

The y-intercept is \(c\), because when \(x = 0\), \(y = c\).

y 轴截距是 \(c\)，因为当 \(x = 0\) 时，\(y = c\)。

A table of values helps you sketch the graph accurately if required.

如果需要，列出若干点的函数值可以帮助你准确画图。

8. Turning Point and Axis of Symmetry | 顶点与对称轴

Every parabola has a turning point (maximum or minimum). The x-coordinate of the turning point is given by:

每条抛物线都有一个顶点（最大值或最小值）。顶点的 x 坐标为：

x = −b / (2a)

The vertical line through \(x = -b/(2a)\) is the axis of symmetry.

穿过 \(x = -b/(2a)\) 的垂直线就是对称轴。

Example: Find the turning point of \(y = x^2 – 4x + 1\).

示例：求 \(y = x^2 – 4x + 1\) 的顶点。

Here \(a = 1\), \(b = -4\), so:

这里 \(a = 1\)，\(b = -4\)，所以：

x = −(−4) / (2×1) = 2

Substitute \(x = 2\) into \(y\):

将 \(x = 2\) 代入 \(y\)：

y = 2² − 4×2 + 1 = 4 − 8 + 1 = −3

Turning point is \((2, -3)\). Since \(a = 1 > 0\), it is a minimum.

顶点是 \((2, -3)\)。因为 \(a = 1 > 0\)，所以是最小值。

9. Applications and Word Problems | 应用与应用题

Quadratic equations often appear in geometry, projectile motion and area problems. Set up an equation from the information given, then solve it.

二次方程经常出现在几何、抛体运动和面积问题中。根据给定信息建立方程，然后求解。

Example: The area of a rectangle is 36 cm². Its length is 5 cm more than its width. Find the width.

示例：一个矩形的面积是 36 cm²。它的长比宽多 5 cm。求宽。

Let width be \(w\). Then length is \(w + 5\).

设宽为 \(w\)。则长为 \(w + 5\)。

w(w + 5) = 36

Expand and rearrange:

展开并整理：

w² + 5w − 36 = 0

Factorise: \((w + 9)(w – 4) = 0\).

因式分解：\((w + 9)(w – 4) = 0\)。

So \(w = -9\) or \(w = 4\). Since width cannot be negative, the width is 4 cm.

所以 \(w = -9\) 或 \(w = 4\)。因为宽不能为负，所以宽为 4 cm。

Check: length = 9 cm, area = 36 cm². Correct.

验证：长为 9 cm，面积为 36 cm²。正确。

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

Many students lose marks on quadratic equations because of small errors. Here are the most common ones:

许多学生在二次方程上失分是因为小错误。以下是最常见的错误：

Forgetting to rearrange the equation to the form \(ax^2 + bx + c = 0\) before factorising or using the formula.
在使用因式分解或公式之前，忘记将方程整理为 \(ax^2 + bx + c = 0\) 的形式。
Making sign errors when substituting negative values into the quadratic formula.
将负值代入二次公式时出现符号错误。
Forgetting that the quadratic formula has a ± symbol, so there can be two solutions.
忘记二次公式中有 ± 符号，所以可能有两个解。
When taking the square root, forgetting to include both positive and negative roots.
开平方时忘记同时取正负根。
Forgetting to check that solutions make sense in word problems (e.g. negative lengths).
在应用题中忘记检查解是否合理（例如长度不能为负）。

Tip: Always write down your working clearly. If you make a small calculator error, the examiner can still award method marks.

提示：始终清晰写下你的步骤。即使计算器有小小失误，考官仍然可以给你方法分。

Practice different types of quadratic questions to build confidence before the exam.

在考试前练习不同类型的二次方程问题以增强信心。

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Solving Quadratic Equations | 解二次方程