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 algebra, graphs, geometry, and problem-solving questions. Mastering the different methods of solving them is essential for exam success.

二次方程是 Edexcel IGCSE 数学大纲中最重要的主题之一。它们出现在代数、图像、几何和解题类题目中。掌握解二次方程的不同方法对考试成功至关重要。

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

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable is 2.

二次方程是指形如 ax² + bx + c = 0 的方程，其中 a、b 和 c 为常数，且 a ≠ 0。变量的最高次数是 2。

For example, x² − 5x + 6 = 0 is a quadratic equation, while x² + 2x + 1 = 0 is another. If a = 0, it becomes a linear equation.

例如，x² − 5x + 6 = 0 是一个二次方程，x² + 2x + 1 = 0 也是。如果 a = 0，它就会变成一个线性方程。

A quadratic equation can have two real roots, one repeated root, or no real roots, depending on the value of the discriminant.

根据判别式的值，二次方程可以有两个实数根、一个重根，或没有实数根。

2. Solving by Factorisation | 因式分解法

Factorisation is often the fastest method when the quadratic expression can be written as a product of two linear factors.

当二次表达式可以写成两个一次因式的乘积时，因式分解通常是最快的方法。

To solve x² + bx + c = 0, find two numbers that multiply to give c and add to give b.

要解 x² + bx + c = 0，找到两个数，它们相乘得 c，相加得 b。

Example: Solve x² − 5x + 6 = 0.

例如：解 x² − 5x + 6 = 0。

We need two numbers whose product is 6 and sum is −5. These are −2 and −3.

我们需要两个数，乘积为 6，和为 −5。这两个数是 −2 和 −3。

(x − 2)(x − 3) = 0

Therefore, x = 2 or x = 3.

因此，x = 2 或 x = 3。

Always check your factors by expanding the brackets.

始终通过展开括号来检查你的因式。

3. Factorising When a ≠ 1 | 当 a ≠ 1 时的因式分解

When the coefficient of x² is not 1, factorisation requires more care.

当 x² 的系数不为 1 时，因式分解需要更加小心。

Consider 2x² + 5x + 2 = 0. We look for two numbers that multiply to a × c = 4 and add to b = 5. These numbers are 1 and 4.

考虑 2x² + 5x + 2 = 0。我们找两个数，它们相乘得 a × c = 4，相加得 b = 5。这两个数是 1 和 4。

Rewrite the middle term: 2x² + 1x + 4x + 2 = 0.

重写中间项：2x² + 1x + 4x + 2 = 0。

Group and factorise: x(2x + 1) + 2(2x + 1) = 0.

分组并提取公因式：x(2x + 1) + 2(2x + 1) = 0。

Now factor out (2x + 1): (2x + 1)(x + 2) = 0.

现在提取 (2x + 1)：(2x + 1)(x + 2) = 0。

This gives x = −1/2 or x = −2.

这得到 x = −1/2 或 x = −2。

Remember: after factoring out the common bracket, the remaining terms form the second bracket.

记住：提取共同的括号后，剩余项组成第二个括号。

4. The Quadratic Formula | 求根公式

The quadratic formula can solve any quadratic equation, including those that cannot be factorised.

求根公式可以解任何二次方程，包括那些无法因式分解的方程。

x = (−b ± √(b² − 4ac)) / (2a)

For ax² + bx + c = 0, substitute the values of a, b and c.

对于 ax² + bx + c = 0，代入 a、b 和 c 的值。

Example: Solve x² − 4x − 7 = 0. Here a = 1, b = −4, c = −7.

例如：解 x² − 4x − 7 = 0。这里 a = 1，b = −4，c = −7。

x = (4 ± √(16 + 28)) / 2 = (4 ± √44) / 2

Since √44 = 2√11, we have x = 2 ± √11.

因为 √44 = 2√11，所以 x = 2 ± √11。

The symbol ± means there are two solutions: one with plus and one with minus.

符号 ± 表示有两个解：一个用加号，一个用减号。

5. Completing the Square | 配方法

Completing the square rewrites x² + bx + c in the form (x + p)² + q.

配方法将 x² + bx + c 改写成 (x + p)² + q 的形式。

Take half of the coefficient of x and square it.

取 x 系数的一半，然后平方。

For x² + 6x + 2 = 0, half of 6 is 3, and 3² = 9.

对于 x² + 6x + 2 = 0，6 的一半是 3，3² = 9。

So x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7.

因此 x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。

Set it equal to zero: (x + 3)² − 7 = 0.

令其等于零：(x + 3)² − 7 = 0。

(x + 3)² = 7

Take the square root of both sides: x + 3 = ±√7, so x = −3 ± √7.

两边开平方：x + 3 = ±√7，所以 x = −3 ± √7。

This method is especially useful for solving equations or finding turning points.

这种方法在解方程或求顶点坐标时特别有用。

6. Solving Quadratic Equations Using Graphs | 用图像解二次方程

The solutions of ax² + bx + c = 0 are the x-coordinates where the graph of y = ax² + bx + c crosses the x-axis.

ax² + bx + c = 0 的解是 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。

If the graph crosses the x-axis at two points, there are two real roots. If it touches the axis at one point, there is one repeated root. If it does not meet the axis, there are no real roots.

如果图像与 x 轴有两个交点，则有两个实数根。如果与轴相切于一点，则有一个重根。如果没有交点，则没有实数根。

Example: The graph of y = x² − 2x − 3 crosses the x-axis at x = −1 and x = 3.

例如：y = x² − 2x − 3 的图像在 x = −1 和 x = 3 处与 x 轴相交。

Therefore, the solutions of x² − 2x − 3 = 0 are x = −1 and x = 3.

因此，x² − 2x − 3 = 0 的解是 x = −1 和 x = 3。

Graphical methods are useful when exact answers are not required, or when the equation is difficult to solve algebraically.

当不需要精确答案，或者方程难以用代数方法求解时，图像法很有用。

7. The Discriminant | 判别式

The discriminant is the expression b² − 4ac inside the square root of the quadratic formula.

判别式是求根公式中根号内的表达式 b² − 4ac。

Its value tells us the nature of the roots without solving the equation.

它的值可以告诉我们根的性质，而无需解方程。

If b² − 4ac > 0, there are two distinct real roots.
若 b² − 4ac > 0，则有两个不同的实数根。
If b² − 4ac = 0, there is exactly one repeated root.
若 b² − 4ac = 0，则恰有一个重根。
If b² − 4ac < 0, there are no real roots.
若 b² − 4ac < 0，则没有实数根。

Example: For 2x² + 3x + 4 = 0, b² − 4ac = 9 − 32 = −23, so there are no real roots.

例如：对于 2x² + 3x + 4 = 0，b² − 4ac = 9 − 32 = −23，所以没有实数根。

The discriminant is also used to determine whether a quadratic can be factorised over integers.

判别式也可用于判断二次式是否能在整数范围内因式分解。

8. Solving Equations by Rearranging | 通过移项解方程

Some equations must be rearranged into standard quadratic form before solving.

有些方程必须先整理成标准二次形式才能求解。

For example, solve x² − 3x = 10.

例如，解 x² − 3x = 10。

Subtract 10 from both sides: x² − 3x − 10 = 0.

两边减去 10：x² − 3x − 10 = 0。

Factorise: (x − 5)(x + 2) = 0, giving x = 5 or x = −2.

因式分解：(x − 5)(x + 2) = 0，得到 x = 5 或 x = −2。

Always move all terms to one side first to get the form ax² + bx + c = 0.

始终先将所有项移到一边，得到 ax² + bx + c = 0 的形式。

Never divide by a variable expression that could be zero; you may lose a root.

切勿除以可能为零的含变量表达式，否则可能会失去一个根。

9. Solving Quadratic Equations with Fractions | 含分数的二次方程

Equations involving algebraic fractions often reduce to quadratics after multiplying through by the denominator.

涉及代数分数的方程通常在乘以分母后化简为二次方程。

Solve 2/x + x/3 = 5 for x, given x ≠ 0.

解 2/x + x/3 = 5，求 x，已知 x ≠ 0。

Multiply both sides by 3x:

两边乘以 3x：

6 + x² = 15x

Rearrange: x² − 15x + 6 = 0.

整理：x² − 15x + 6 = 0。

This does not factorise simply, so use the quadratic formula.

这个不能简单因式分解，所以使用求根公式。

x = (15 ± √(225 − 24)) / 2 = (15 ± √201) / 2

Always check that your final solutions do not make any original denominator zero.

始终检查最终解是否会使原分母为零。

10. Solving Quadratic Inequalities | 二次不等式

Quadratic inequalities are solved by first solving the corresponding quadratic equation, then testing intervals.

解二次不等式时，先求解对应的二次方程，然后检验区间。

Solve x² − 4x + 3 > 0.

解 x² − 4x + 3 > 0。

First factorise: (x − 1)(x − 3) > 0.

先因式分解：(x − 1)(x − 3) > 0。

The roots are x = 1 and x = 3. These split the number line into three regions.

根是 x = 1 和 x = 3。它们将数轴分成三个区间。

For x < 1, both factors are negative, product is positive.
当 x < 1 时，两个因子都为负，乘积为正。
For 1 < x < 3, one factor is negative, one positive, product is negative.
当 1 < x < 3 时，一个因子为负，另一个为正，乘积为负。
For x > 3, both factors are positive, product is positive.
当 x > 3 时，两个因子都为正，乘积为正。

Therefore the solution is x < 1 or x > 3.

因此解是 x < 1 或 x > 3。

Remember to use ≤ or ≥ if the inequality includes equality.

如果不等式包含等号，记得使用 ≤ 或 ≥。

11. Word Problems from Common Exam Contexts | 常见考试情境中的应用题

Quadratic equations frequently appear in geometry, area, and motion problems.

二次方程经常出现在几何、面积和运动问题中。

Example: A rectangle has length 2 cm longer than its width. Its area is 24 cm². Find the width.

例：一个长方形的长比宽长 2 厘米。它的面积是 24 平方厘米。求宽。

Let the width be x cm. Then the length is (x + 2) cm.

设宽为 x 厘米。则长为 (x + 2) 厘米。

Area equation: x(x + 2) = 24.

面积方程：x(x + 2) = 24。

Expanding and rearranging: x² + 2x − 24 = 0.

展开并整理：x² + 2x − 24 = 0。

Factorise: (x + 6)(x − 4) = 0, so x = 4 or x = −6.

因式分解：(x + 6)(x − 4) = 0，所以 x = 4 或 x = −6。

Since width cannot be negative, x = 4 cm.

因为宽度不能为负，所以 x = 4 厘米。

Always state the answer with appropriate units and reject invalid solutions.

始终使用适当的单位给出答案，并舍弃无效的解。

12. Common Exam Tips and Mistakes | 考试常见技巧与错误

Here are important tips to avoid losing marks in the exam.

以下是一些避免在考试中失分的重要技巧。

Always rearrange into the form ax² + bx + c = 0 before factorising or using the formula.
在因式分解或使用公式前，始终将方程整理成 ax² + bx + c = 0 的形式。
When using the quadratic formula, check that your value of a is the coefficient of x², not x.
使用求根公式时，检查 a 是 x² 的系数，而不是 x 的系数。
When factorising, expand your answer to verify that it matches the original expression.
因式分解后，展开你的答案以验证它是否与原式一致。
Give exact answers unless the question asks for a decimal approximation.
除非题目要求小数近似值，否则给出精确答案。
For word problems, define the variable clearly and include units in your final answer.
对于应用题，清晰地定义变量，并在最终答案中包含单位。
If an equation cannot be factorised, do not guess — use the formula or complete the square.
如果方程无法因式分解，不要猜测，使用求根公式或配方法。

Practice different types of quadratic equations until the methods become automatic.

练习不同类型的二次方程，直到这些方法变得熟练自如。

Summary of Methods | 方法总结

Method	Best Used When	Example
Factorisation	The expression factors easily	x² − 5x + 6 = 0
Quadratic formula	Any quadratic, especially non-factorisable	x² + 2x − 5 = 0
Completing the square	Finding turning points or solving exactly	x² + 6x + 2 = 0
Graphing	Approximate solutions or checking answers	y = x² − 2x − 3

Choose the most efficient method for the equation you face. In the exam, if factorisation is not obvious, use the quadratic formula or complete the square.

根据你所面对的方程选择最有效的方法。在考试中，如果因式分解不明显，就使用求根公式或配方法。

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Solving Quadratic Equations | 解二次方程