Solving Quadratic Equations | 解一元二次方程

📚 Solving Quadratic Equations | 解一元二次方程

Quadratic equations appear in almost every IGCSE Mathematics paper. Mastering the methods of solving them is essential for success in the Edexcel syllabus.

二次方程几乎出现在每一份 IGCSE 数学试卷中。掌握解二次方程的方法对于在 Edexcel 考纲中取得好成绩至关重要。

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

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数，且 a ≠ 0。

The highest power of x is 2, which gives the equation its name “quadratic”.

x 的最高次数是 2，因此得名“二次”。

If a = 0, the equation becomes linear, so a must not be zero.

若 a = 0，方程变为一次方程，因此 a 不能为零。

2. Solving by Factorisation | 因式分解法

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

当二次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。

If ab = 0, then a = 0 or b = 0.

若 ab = 0，则 a = 0 或 b = 0。

Example: Solve x² − 5x + 6 = 0. Factorise to (x − 2)(x − 3) = 0. Hence x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。因式分解得 (x − 2)(x − 3) = 0，因此 x = 2 或 x = 3。

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

When the coefficient of x² is not 1, we need to find two numbers whose product is ac and whose sum is b.

当 x² 的系数不为 1 时，我们需要找到两个数，它们的乘积为 ac，和为 b。

Example: Solve 2x² + 7x + 3 = 0. Here ac = 6 and b = 7. The numbers 1 and 6 work. Write 2x² + x + 6x + 3 = 0, then factor by grouping: x(2x + 1) + 3(2x + 1) = 0, giving (2x + 1)(x + 3) = 0.

例：解 2x² + 7x + 3 = 0。这里 ac = 6，b = 7。数字 1 和 6 满足条件。写成 2x² + x + 6x + 3 = 0，然后分组因式分解：x(2x + 1) + 3(2x + 1) = 0，得到 (2x + 1)(x + 3) = 0。

Hence x = −½ or x = −3.

因此 x = −½ 或 x = −3。

4. Solving by Completing the Square | 配方法

Completing the square rewrites the quadratic in the form a(x + p)² + q. This is especially useful when factorisation is not possible.

配方法将二次式改写为 a(x + p)² + q 的形式。当无法因式分解时，这种方法尤其有用。

x² + bx = (x + b/2)² − (b/2)²

Example: Solve x² + 6x − 1 = 0. Complete the square: (x + 3)² − 9 − 1 = 0, so (x + 3)² = 10. Therefore x = −3 ± √10.

例：解 x² + 6x − 1 = 0。配方得 (x + 3)² − 9 − 1 = 0，所以 (x + 3)² = 10。因此 x = −3 ± √10。

5. The Quadratic Formula | 求根公式

The quadratic formula works for all quadratic equations, provided a ≠ 0. It is derived from completing the square.

求根公式适用于所有二次方程，只要 a ≠ 0。它由配方法推导而来。

x = (−b ± √(b² − 4ac)) / (2a)

Example: Solve 2x² − 4x − 3 = 0 using the formula. Here a = 2, b = −4, c = −3. Substitute: x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = 1 ± (√10)/2.

例：用公式解 2x² − 4x − 3 = 0。这里 a = 2，b = −4，c = −3。代入得 x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 = 1 ± (√10)/2。

6. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us how many real roots a quadratic equation has.

表达式 b² − 4ac 称为判别式。它告诉我们二次方程有几个实数根。

Discriminant Δ	Number of real roots	实数根个数
Δ > 0	Two distinct roots	两个不相等的实数根
Δ = 0	One repeated root	一个重根
Δ < 0	No real roots	没有实数根

7. Roots and Coefficients | 根与系数的关系

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of the roots is −b/a and the product is c/a.

对于二次方程 ax² + bx + c = 0，若其根为 α 和 β，则两根之和为 −b/a，两根之积为 c/a。

α + β = −b/a, αβ = c/a

This relationship is handy for checking solutions or constructing equations with given roots.

这个关系可用于检验解，或构造具有给定根的方程。

8. Quadratic Equations in Context | 二次方程的应用题

Many word problems lead to quadratic equations. Common contexts include area, projectile motion and number problems.

许多文字应用题都会归结到二次方程。常见情境包括面积、抛体运动和数字问题。

Example: A rectangle has length (x + 3) cm and width (x − 1) cm. Its area is 12 cm². Find x.

例：一个长方形的长为 (x + 3) cm，宽为 (x − 1) cm，面积为 12 cm²。求 x。

Set up: (x + 3)(x − 1) = 12 → x² + 2x − 3 = 12 → x² + 2x − 15 = 0 → (x + 5)(x − 3) = 0. Since x > 0, x = 3.

列式：(x + 3)(x − 1) = 12 → x² + 2x − 3 = 12 → x² + 2x − 15 = 0 → (x + 5)(x − 3) = 0。因为 x > 0，所以 x = 3。

9. Choosing the Best Method | 选择最佳解法

On the exam, you need to decide which method is most efficient. Here is a quick guide.

在考试中，你需要判断哪种方法最有效。以下是一个快速指南。

If the equation factorises easily, use factorisation.

如果可以轻松因式分解，请用因式分解法。

If the coefficient of x² is 1 and b is even, completing the square may be fast.

如果 x² 系数为 1 且 b 为偶数，配方法可能很快。

If the question asks for answers to a given degree of accuracy, use the quadratic formula.

如果题目要求答案保留到指定精度，请使用求根公式。

10. Common Mistakes | 常见错误

Avoid these frequent errors to gain more marks in your IGCSE exam.

避免以下常见错误，以便在 IGCSE 考试中拿到更多分数。

Forgetting that ax² + bx + c = 0 must be rearranged to zero before factorising or using the formula.

忘记在使用因式分解或公式前，需要将方程整理成 ax² + bx + c = 0 的形式。

Losing the negative sign when substituting into the quadratic formula.

在代入求根公式时丢失负号。

Giving only one solution when the equation has two roots.

当方程有两个根时只给出一个解。

11. Practice Tip | 练习建议

Try solving the same equation using two different methods. If you get the same answers, you are likely correct.

尝试用两种不同方法解同一个方程。如果得到相同答案，你很可能做对了。

Example: x² − 4x − 12 = 0. Factorisation gives (x − 6)(x + 2) = 0, so x = 6 or x = −2. The formula gives the same results.

例：x² − 4x − 12 = 0。因式分解得到 (x − 6)(x + 2) = 0，所以 x = 6 或 x = −2。公式法也得到相同结果。

12. Summary | 总结

Quadratic equations are a core topic in IGCSE Mathematics. You must be comfortable with factorisation, completing the square and the quadratic formula. Understand the discriminant and apply these skills to word problems.

二次方程是 IGCSE 数学的核心内容。你必须熟练掌握因式分解法、配方法和求根公式。理解判别式的含义，并将这些技能应用于文字应用题。

Practise past paper questions regularly to build confidence and speed.

定期练习历年真题，可以增强信心并提高速度。

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Solving Quadratic Equations | 解一元二次方程