Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear in almost every IGCSE Mathematics exam. Mastering the different methods of solving them is essential for success. This revision guide from TutorHao covers everything you need, from factorisation to the quadratic formula, with clear examples and exam tips.

二次方程几乎出现在每一场 IGCSE 数学考试中。掌握不同的解题方法对取得好成绩至关重要。本 TutorHao 复习指南涵盖了你需要的一切，从因式分解到求根公式，配有清晰的例题和考试技巧。

1. Standard Form and Key Terms | 标准形式与关键术语

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. The highest power of the variable is 2.

二次方程是任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数且 a ≠ 0。变量的最高次数是 2。

Roots / Solutions: The values of x that make the equation true.
解 / 根: 使方程成立的 x 的值。
Coefficient: The number multiplying the variable, e.g. 3 in 3x².
系数: 与变量相乘的数，例如 3x² 中的 3。
Constant: The term without a variable.
常数项: 不含变量的项。

Always rearrange the equation into the standard form before attempting to solve it.

在尝试求解之前，务必先将方程整理成标准形式。

2. Solving by Factorisation | 因式分解法

If a quadratic expression can be factorised into two brackets, then we can use the zero product property: if p × q = 0, then p = 0 or q = 0.

如果一个二次表达式可以分解成两个括号相乘，那么我们可以利用零积性质：若 p × q = 0，则 p = 0 或 q = 0。

Example: Solve x² – 5x + 6 = 0.

例：解 x² – 5x + 6 = 0。

(x – 2)(x – 3) = 0

所以 x = 2 或 x = 3

Check that the brackets expand correctly: x² – 3x – 2x + 6 = x² – 5x + 6. Always expand back to verify.

检查括号展开是否正确：x² – 3x – 2x + 6 = x² – 5x + 6。务必展开验证。

3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic as a(x + p)² + q. This method works for any quadratic, even when factorisation is difficult.

配方法将二次式改写为 a(x + p)² + q 的形式。这种方法适用于任何二次方程，即使难以因式分解。

Example: Solve x² + 6x – 7 = 0 by completing the square.

例：用配方法解 x² + 6x – 7 = 0。

(x + 3)² – 9 – 7 = 0

(x + 3)² = 16

x + 3 = ±4

x = 1 或 x = –7

Remember: when the coefficient of x² is not 1, first factor out a from the x² and x terms.

记住：当 x² 的系数不为 1 时，先提取 a 到括号外。

4. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation of the form ax² + bx + c = 0. It is essential to memorise it.

求根公式可以解任何形式为 ax² + bx + c = 0 的二次方程。记住它是必不可少的。

x = (–b ± √(b² – 4ac)) / (2a)

Example: Solve 2x² + 3x – 5 = 0 using the formula.

例：用公式解 2x² + 3x – 5 = 0。

Here a = 2, b = 3, c = –5. Substitute into the formula:

这里 a = 2，b = 3，c = –5。代入公式：

x = (–3 ± √(9 + 40)) / 4 = (–3 ± 7) / 4

x = 1 或 x = –5/2

Be careful with negative signs when substituting c.

代入 c 时注意负号。

5. 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多少个实数根。

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

This is often tested in questions about the nature of roots. It also helps you decide which method to use.

这在关于根的性质的题目中经常考查。它还能帮助你决定使用哪种方法。

6. Graphs of Quadratic Functions | 二次函数图像

The graph of y = ax² + bx + c is a parabola. If a > 0 it opens upwards (a U shape). If a < 0 it opens downwards (an n shape).

y = ax² + bx + c 的图像是一条抛物线。若 a > 0，开口向上（U 形）。若 a < 0，开口向下（n 形）。

The roots of the equation ax² + bx + c = 0 are the x-intercepts of the graph. The y-intercept is always c.

方程 ax² + bx + c = 0 的根就是图像与 x 轴的交点。y 轴截距始终是 c。

If the graph touches the x-axis at one point, the equation has one repeated root.
如果图像与 x 轴只有一个交点，则方程有一个重根。
If the graph does not cross the x-axis, there are no real roots.
如果图像不与 x 轴相交，则没有实数根。

Sketching graphs therefore gives a visual meaning to the discriminant.

因此画图可以直观地理解判别式。

7. Roots, Intercepts and Turning Points | 根、截距与顶点

The turning point (vertex) of a parabola can be found by completing the square. In the form y = a(x – h)² + k, the vertex is at (h, k).

抛物线的顶点可以通过配方求得。在 y = a(x – h)² + k 的形式中，顶点为 (h, k)。

Alternatively, the x-coordinate of the vertex is the midpoint of the two roots, or –b/(2a).

另外，顶点的 x 坐标是两个根的中点，即 –b/(2a)。

Example: Find the turning point of y = x² – 4x + 5.

例：求 y = x² – 4x + 5 的顶点。

y = (x – 2)² – 4 + 5 = (x – 2)² + 1

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

所以顶点为 (2, 1)。由于 a = 1 > 0，这是最小值点。

8. Quadratic Inequalities | 二次不等式

Solving a quadratic inequality involves finding the range of x values for which the quadratic is positive or negative.

解二次不等式需要找出使二次式为正值或负值的 x 的取值范围。

Example: Solve x² – x – 6 > 0.

例：解 x² – x – 6 > 0。

First factorise: (x – 3)(x + 2) > 0. The roots are x = –2 and x = 3.

先因式分解：(x – 3)(x + 2) > 0。根为 x = –2 和 x = 3。

Sketch the graph: it opens upwards, so it is positive outside the roots.

画出草图：开口向上，所以在两根之外取正值。

x < –2 或 x > 3

Remember to change the inequality sign when multiplying or dividing by a negative number.

记住：乘以或除以负数时要改变不等号方向。

9. Modelling with Quadratics | 二次方程建模

Many real-world problems can be modelled by quadratic equations, such as projectile motion, area optimisation, and profit functions.

许多现实问题可以用二次方程建模，例如抛体运动、面积优化和利润函数。

Example: A rectangle has length (x + 3) cm and width (x – 1) cm. Its area is 45 cm². Find x.

例：一个矩形的长为 (x + 3) cm，宽为 (x – 1) cm，面积为 45 cm²。求 x。

(x + 3)(x – 1) = 45

x² + 2x – 3 = 45

x² + 2x – 48 = 0

(x + 8)(x – 6) = 0

Since x represents a length, x = 6 (discard x = –8).

由于 x 表示长度，所以 x = 6（舍去 x = –8）。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Students often make avoidable errors in quadratic questions. Here are the most common ones and how to avoid them.

学生在二次方程题目中经常犯一些可以避免的错误。以下是最常见的错误以及如何避免。

Not rearranging to zero: Always set the equation as ax² + bx + c = 0 before factorising or applying the formula.
未整理成零的形式: 在因式分解或套用公式之前，务必把方程写成 ax² + bx + c = 0。
Sign errors: Double-check signs when substituting into the quadratic formula, especially –c.
符号错误: 代入求根公式时仔细检查符号，尤其是 –c。
Missing the ± in the formula: The quadratic formula gives two solutions; do not forget the ±.
漏掉公式中的 ±: 求根公式给出两个解；不要忘记 ±。
Forgetting to expand and check: Always expand your factorised answer to confirm it matches the original.
忘记展开检查: 始终展开因式分解的结果，确认与原式一致。

In the exam, show all steps clearly. Even if your final answer is wrong, you may earn method marks.

考试中要清晰展示每一步。即使最终答案错误，也可能获得方法分。

11. Practice Questions | 练习题

Try these exam-style questions to test your understanding. Full solutions are included below.

试试以下考试风格题目来测试你的理解。完整解答附在后面。

1. Solve x² – 7x + 10 = 0

2. Solve 3x² + 5x – 2 = 0

3. Find the discriminant of 2x² – 4x + 3

4. Solve x² – 6x + 1 = 0 by completing the square

Answers:

答案：

1. (x – 2)(x – 5) = 0 → x = 2 or x = 5
1. (x – 2)(x – 5) = 0 → x = 2 或 x = 5
2. (3x – 1)(x + 2) = 0 → x = 1/3 or x = –2
2. (3x – 1)(x + 2) = 0 → x = 1/3 或 x = –2
3. Δ = (–4)² – 4(2)(3) = 16 – 24 = –8
3. Δ = (–4)² – 4(2)(3) = 16 – 24 = –8
4. (x – 3)² – 8 = 0 → x = 3 ± √8 = 3 ± 2√2
4. (x – 3)² – 8 = 0 → x = 3 ± √8 = 3 ± 2√2

Make sure you can solve each type without looking at the answers first.

确保你能不先看答案就解出每类题目。

12. Summary | 总结

Quadratic equations are a central topic in IGCSE Edexcel Mathematics. You should be able to solve them by factorisation, completing the square, or using the quadratic formula. Understand the discriminant, and be confident with interpreting graphs and turning points.

二次方程是 IGCSE Edexcel 数学的核心主题。你应该能够通过因式分解、配方法或求根公式来求解它们。理解判别式，并能熟练解读图像和顶点。

Keep practising past papers and check your working step by step. With consistent revision, this topic can become an easy scoring area for you.

不断练习真题，并逐步检查你的解题过程。持续复习，这个主题可以成为你容易得分的部分。

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Solving Quadratic Equations | 解二次方程