Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

Quadratic equations appear frequently in IGCSE Mathematics. This revision guide explains the main methods for solving them, highlights common pitfalls, and provides structured practice to help you secure full marks in your exam.

二次方程在 IGCSE 数学中频繁出现。本复习指南讲解其主要的解法，指出常见易错点，并提供结构化练习，帮助你在考试中拿满分数。

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

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

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是实数，且 a ≠ 0。

The highest power of the variable is 2, which is why the graph of a quadratic is a parabola. Solving a quadratic means finding the values of x that make the equation true.

变量的最高次数是 2，因此二次函数的图像是抛物线。解二次方程就是求出使等式成立的 x 的值。

2. Solving by Factorisation | 因式分解法

If the quadratic expression can be factorised into two brackets, then the equation can be solved by setting each bracket equal to zero.

如果二次式能分解成两个括号相乘的形式，那么可以通过令每个括号等于零来求解方程。

For example, solve x² – 5x + 6 = 0. Factorise to get (x – 2)(x – 3) = 0. Then x – 2 = 0 or x – 3 = 0, so x = 2 or x = 3.

例如，解 x² – 5x + 6 = 0。因式分解得 (x – 2)(x – 3) = 0。于是 x – 2 = 0 或 x – 3 = 0，所以 x = 2 或 x = 3。

This method works best when a = 1 and the constant term has simple factors. For harder cases, the quadratic formula is more reliable.

当 a = 1 且常数项因子简单时，这种方法最有效。如果情况较复杂，使用求根公式更可靠。

3. Solving by the Quadratic Formula | 公式法

The quadratic formula gives the solution to any quadratic equation ax² + bx + c = 0:

二次求根公式给出了任意二次方程 ax² + bx + c = 0 的解：

x = (−b ± √(b² − 4ac)) / (2a)

To use it, substitute the values of a, b and c carefully. Pay attention to signs because a mistake here changes the answer completely.

使用时，要小心代入 a、b、c 的值。特别注意符号，因为这里的错误会导致答案完全改变。

Example: solve 2x² – 4x – 3 = 0. Here a = 2, b = −4, c = −3. Substituting gives x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4. So x ≈ 2.58 or x ≈ −0.58.

例：解 2x² – 4x – 3 = 0。这里 a = 2，b = −4，c = −3。代入得 x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4。所以 x ≈ 2.58 或 x ≈ −0.58。

4. Solving by Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This is useful for solving equations and also for finding the turning point of a quadratic graph.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这种方法既可用于解方程，也用于求二次图像的顶点。

The key step is to take half of the coefficient of x, square it, and adjust the constant term.

关键步骤是取 x 系数的一半，平方，然后调整常数项。

For example, solve x² + 6x + 4 = 0. Half of 6 is 3, and 3² = 9. So x² + 6x + 4 = (x + 3)² – 9 + 4 = (x + 3)² – 5 = 0. Hence (x + 3)² = 5, so x = −3 ± √5.

例如，解 x² + 6x + 4 = 0。6 的一半是 3，3² = 9。所以 x² + 6x + 4 = (x + 3)² – 9 + 4 = (x + 3)² – 5 = 0。于是 (x + 3)² = 5，所以 x = −3 ± √5。

5. The Discriminant | 判别式

The discriminant of a quadratic equation is the expression Δ = b² – 4ac. It tells us how many real roots the equation has.

二次方程的判别式是 Δ = b² – 4ac。它告诉我们方程有多少个实数根。

If Δ > 0, there are two distinct real roots. If Δ = 0, there is exactly one repeated root. If Δ < 0, there are no real roots.

如果 Δ > 0，有两个不同的实数根；如果 Δ = 0，有一个重根；如果 Δ < 0，没有实数根。

Discriminant	Number of real roots
b² – 4ac > 0	Two roots
b² – 4ac = 0	One repeated root
b² – 4ac < 0	No real roots

Working with the discriminant alone also appears in questions that ask whether a line intersects, is tangent to, or misses a quadratic curve.

只使用判别式也出现在这样的题目中：判断一条直线与一条二次曲线是相交、相切还是相离。

6. Solving Word Problems | 应用题解题

Many exam problems describe a real situation and require you to form and solve a quadratic equation. Read carefully for phrases like “product is”, “area is” or “the square of a number”.

许多考试题目描述一个实际情境，要求你建立并解一个二次方程。仔细阅读“积为”“面积为”或“某数的平方”等短语。

Example: the length of a rectangle is 4 cm longer than its width, and the area is 12 cm². Let width be x. Then length = x + 4, so x(x + 4) = 12. Expanding gives x² + 4x – 12 = 0. Factorising: (x + 6)(x – 2) = 0. Since width cannot be negative, x = 2.

例：一个长方形的长比宽长 4 cm，面积为 12 cm²。设宽为 x，则长为 x + 4，所以 x(x + 4) = 12。展开得 x² + 4x – 12 = 0。因式分解：(x + 6)(x – 2) = 0。由于宽不能为负数，所以 x = 2。

Always check whether your solutions make sense in the context of the problem, and reject negative or impossible answers.

始终检查解在问题情境中是否有意义，并舍弃负根或不可能的答案。

7. Common Mistakes | 常见错误

One common mistake is forgetting to rearrange the equation to zero before factorising. For example, x² = 4x must be rewritten as x² – 4x = 0 before you solve it.

一个常见错误是在因式分解前忘记把方程化为等于零的形式。例如，x² = 4x 必须改写成 x² – 4x = 0 再求解。

Another mistake is dividing both sides of an equation by x, incorrectly losing a root. You can only divide by a quantity if you are sure it is not zero.

另一个错误是两边同时除以 x，导致丢失一个根。只有当你确定这个量不为零时，才能除以它。

Also, when using the quadratic formula, check that negative signs are included inside the square root. Writing b² – 4ac instead of (−b)² – 4ac is a frequent slip when b is negative.

另外，使用求根公式时，注意负号要包含在平方根内。当 b 为负数时，把 b² – 4ac 写成 (−b)² – 4ac 是常见的笔误。

8. Practice Questions | 练习题目

Try these short questions to test your understanding. Do not use a calculator for the first three.

尝试下列小题来检验你的理解。前三题请不要用计算器。

Solve x² = 9.
Solve x² – 7x + 10 = 0.
Solve x² + 2x – 8 = 0.
Use the quadratic formula to solve 3x² + 5x – 2 = 0.
Find the value of the discriminant of x² – 4x + 4 = 0.

Answers: 1) x = ±3. 2) x = 2 or 5. 3) x = −4 or 2. 4) x = ⅓ or −2. 5) Δ = 0.

答案：1) x = ±3；2) x = 2 或 5；3) x = −4 或 2；4) x = ⅓ 或 −2；5) Δ = 0。

9. Graphical Interpretation | 图像意义

The solutions of a quadratic equation are the x-coordinates where the graph of y = ax² + bx + c crosses the x-axis. If the graph touches the axis at one point, there is one repeated root; if it never crosses, the equation has no real roots.

二次方程的解就是 y = ax² + bx + c 的图像与 x 轴交点的横坐标。如果图像在一点与轴相切，则有一个重根；如果永远不相交，则方程没有实数根。

This idea links algebra and geometry, and it is often tested in questions that ask you to sketch a graph or interpret a given graph.

这个思想将代数与几何联系起来，常在要求你画出草图或解读给定图像的问题中考查。

10. Final Summary | 总结

You should be able to solve a quadratic equation by factorisation, completing the square, or using the quadratic formula. Also check the discriminant when asked about the number of roots.

你应该能够用因式分解法、配方法或求根公式求解二次方程。当被问及根的数目时，应使用判别式。

Always simplify your answers where possible and present them clearly. With regular practice, quadratic equations become one of the most reliable sections in the IGCSE mathematics paper.

在可能的情况下始终化简答案，并清晰书写。通过有规律的练习，二次方程会成为 IGCSE 数学试卷中最有把握拿分的板块之一。

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Solving Quadratic Equations | 解二次方程