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 IGCSE Mathematics. They appear in algebra, graphs, geometry, and even in problem-solving questions. Mastering the different methods of solving quadratics is essential for exam success.

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、几何甚至应用题中。掌握解二次方程的不同方法是考试成功的关键。

1. Standard Form of a Quadratic Equation | 二次方程的标准形式

A quadratic equation is an equation where the highest power of the variable is 2. The standard form is written as:

二次方程是指变量最高次数为 2 的方程。其标准形式为：

ax² + bx + c = 0

where a, b and c are constants, and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.

其中 a、b、c 为常数，且 a ≠ 0。如果 a = 0，方程就变成一次方程，而不是二次方程。

Example: 2x² − 3x + 1 = 0 is quadratic, but 2x − 3 = 0 is not.
例：2x² − 3x + 1 = 0 是二次方程，而 2x − 3 = 0 不是。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic can be factored easily. You look for two numbers that multiply to give ac and add to give b.

当二次式容易分解时，因式分解法通常是最快的方法。你要找到两个数，它们的乘积等于 ac，和等于 b。

For the equation x² + 5x + 6 = 0, we need two numbers that multiply to 6 and add to 5. The numbers are 2 and 3.

对于方程 x² + 5x + 6 = 0，我们需要找到两个数，乘积为 6，和为 5。这两个数是 2 和 3。

(x + 2)(x + 3) = 0

Then set each bracket equal to zero: x + 2 = 0 or x + 3 = 0, giving x = −2 or x = −3.

然后令每个括号等于零：x + 2 = 0 或 x + 3 = 0，得到 x = −2 或 x = −3。

Always check if the coefficient of x² is 1 or not before using simple factorisation.
在使用简单因式分解之前，先检查 x² 的系数是否为 1。
If the coefficient is not 1, use the ac method or trial and error.
如果系数不是 1，使用 ac 法或试错法。

3. The Quadratic Formula | 求根公式

When factorisation is difficult or impossible, the quadratic formula always works. For ax² + bx + c = 0, the solutions are:

当因式分解困难或不可能时，求根公式总是有效的。对于 ax² + bx + c = 0，解为：

x = (−b ± √(b² − 4ac)) / (2a)

This formula gives two solutions, one with the plus sign and one with the minus sign. You must memorise this formula for the exam.

这个公式给出两个解，一个取加号，一个取减号。你必须在考试中记住这个公式。

Example: Solve 2x² + 3x − 2 = 0 using the formula.
例：使用公式解 2x² + 3x − 2 = 0。
Here a = 2, b = 3, c = −2. Substitute into the formula.
这里 a = 2，b = 3，c = −2。代入公式。

x = (−3 ± √(3² − 4×2×(−2))) / (2×2) = (−3 ± √(9 + 16)) / 4 = (−3 ± 5) / 4

So x = (−3 + 5)/4 = 0.5 or x = (−3 − 5)/4 = −2.

因此 x = (−3 + 5)/4 = 0.5 或 x = (−3 − 5)/4 = −2。

4. Completing the Square | 配方法

Completing the square is another method that also helps you find the turning point of a quadratic graph. You rewrite the quadratic in the form a(x + p)² + q.

配方法是另一种解法，它还能帮助你找到二次图像的顶点。你将二次式改写为 a(x + p)² + q 的形式。

For x² + 6x + 5 = 0, take half of 6, which is 3, and square it to get 9. Then:

对于 x² + 6x + 5 = 0，取 6 的一半是 3，平方得到 9。于是：

(x + 3)² − 9 + 5 = 0

(x + 3)² − 4 = 0

Then (x + 3)² = 4, so x + 3 = ±2, giving x = −1 or x = −5.

然后 (x + 3)² = 4，所以 x + 3 = ±2，得到 x = −1 或 x = −5。

If the coefficient of x² is not 1, factor it out first before completing the square.
如果 x² 的系数不是 1，先提出来再进行配方。

5. Solving Quadratic Equations by Graph | 图像法解二次方程

You can also solve a quadratic equation by drawing its graph. The solutions are the x-coordinates where the curve crosses the x-axis, i.e. where y = 0.

你也可以通过画二次函数图像来解二次方程。解是曲线与 x 轴交点的 x 坐标，即 y = 0 的位置。

If the graph crosses the x-axis at two points, there are two real roots.
如果图像与 x 轴有两个交点，则有两个实根。
If the graph touches the x-axis at one point, there is one repeated root.
如果图像与 x 轴只有一个切点，则有一个重根。
If the graph does not touch the x-axis, there are no real roots.
如果图像不与 x 轴相交，则没有实根。

Graphical methods are useful for estimating solutions, but algebraic methods give exact answers.

图像法适合估计解，但代数方法给出精确答案。

6. The Discriminant | 判别式

The discriminant is the part of the quadratic formula under the square root: b² − 4ac. It tells you the nature of the roots without solving the equation.

判别式是求根公式中根号下的部分：b² − 4ac。它无需解方程就能判断根的性质。

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

例：对于 x² − 4x + 4 = 0，判别式 = (−4)² − 4×1×4 = 16 − 16 = 0，所以有一个重根。

Example: For x² − 4x + 4 = 0, the discriminant = (−4)² − 4×1×4 = 16 − 16 = 0, so there is one repeated root.

7. Forming Quadratic Equations from Roots | 由根构造二次方程

If you know the roots of a quadratic equation, say α and β, then the equation can be written as:

如果你知道二次方程的根，设为 α 和 β，则方程可以写成：

x² − (α + β)x + αβ = 0

This is a useful shortcut for problems that give you the sum and product of roots.

这是一个有用的技巧，适用于给出根的和与积的题目。

Sum of roots = −b/a
根的和 = −b/a
Product of roots = c/a
根的积 = c/a

8. Applications in Problem Solving | 二次方程的应用题

Many IGCSE problems require you to set up a quadratic equation from a real-life situation. Common contexts include area, projectile motion, and number problems.

许多 IGCSE 题目要求你从实际情境中建立二次方程。常见情境包括面积、抛体运动和数字问题。

Read the question carefully and define a variable.
仔细阅读题目并定义一个变量。
Translate the information into an equation.
将信息转化为方程。
Solve the equation using any valid method.
用任意有效方法解方程。
Check whether both solutions make sense in the context. Discard any negative length or time.
检查两个解是否在情境中合理。舍弃负长度或负时间等不合理的解。

Example: A rectangle has length 3 cm longer than its width. Its area is 40 cm². Let width = x. Then length = x + 3. So x(x + 3) = 40, which gives x² + 3x − 40 = 0.

例：一个矩形的长比宽长 3 cm，面积为 40 cm²。设宽为 x，则长为 x + 3。所以 x(x + 3) = 40，即 x² + 3x − 40 = 0。

x² + 3x − 40 = 0 → (x + 8)(x − 5) = 0

So x = −8 or x = 5. Since width cannot be negative, x = 5 cm and length = 8 cm.

所以 x = −8 或 x = 5。因为宽不能为负，所以 x = 5 cm，长为 8 cm。

9. Common Mistakes to Avoid | 常见错误提醒

Students often lose marks due to small errors. Here are the most common ones:

学生常因小错误失分。以下是最常见的错误：

Forgetting to write the equation in the form ax² + bx + c = 0 before factorising.
在因式分解前忘记将方程写成 ax² + bx + c = 0 的形式。
Using the quadratic formula with the wrong sign for c.
使用求根公式时把 c 的符号弄错。
Ignoring the ± sign, which gives only one solution.
忽略 ± 号，只得到一个解。
When solving word problems, not rejecting a negative solution that is impossible.
解应用题时，没有舍去不可能的负数解。

10. Exam Tips | 考试技巧

In the IGCSE Edexcel exam, you should choose the most efficient method for each question. Show all working clearly, as method marks are awarded.

在 IGCSE Edexcel 考试中，你应该为每道题选择最有效的方法。清楚展示所有步骤，因为步骤分很重要。

If the question says “give your answer correct to 2 decimal places”, use the quadratic formula and round at the end.
如果题目要求”答案精确到两位小数”，应使用求根公式并在最后四舍五入。
Factorisation is usually easier for integer answers.
当答案是整数时，因式分解通常更简单。
Completing the square is useful when you also need the turning point of the graph.
当你还需要图像的顶点时，配方法很有用。

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Solving Quadratic Equations | 解二次方程