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 standard methods of solving them is essential for the Edexcel examination.

二次方程几乎出现在每一份 IGCSE 数学试卷中。掌握解二次方程的标准方法是 Edexcel 考试取得成功的关键。

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

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 x is 2.

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

For example, 2x² – 5x + 3 = 0 is a quadratic equation. If an equation has an x² term and no higher powers, it can usually be rearranged into this standard form.

例如，2x² – 5x + 3 = 0 是一个二次方程。若一个方程含有 x² 项且没有更高次项，通常可整理成这种标准形式。

The coefficient a is the number before x².
The coefficient b is the number before x.
The constant term is c.

系数 a 是 x² 前面的数。
系数 b 是 x 前面的数。
常数项是 c。

2. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic expression factorises neatly. The idea is to write the equation as a product of two linear factors equal to zero.

当二次表达式可以整齐地分解时，因式分解法通常是最快的方法。其核心是把方程写成两个一次因式的乘积等于零的形式。

For example, solve x² – 5x + 6 = 0. We look for two numbers that multiply to 6 and add to -5. These are -2 and -3, so the factorised form is (x – 2)(x – 3) = 0.

例如，解 x² – 5x + 6 = 0。我们寻找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3，因此因式分解为 (x – 2)(x – 3) = 0。

(x – 2)(x – 3) = 0 → x – 2 = 0 or x – 3 = 0 → x = 2 or x = 3

If the product of two expressions is zero, at least one of them must be zero. This is called the zero product property.

如果两个表达式的乘积为零，则至少其中一个必须为零。这称为零乘积性质。

3. Factorising More Complex Quadratics | 分解更复杂的二次式

When the coefficient a is not 1, factorisation requires extra care. For example, solve 2x² + 7x + 3 = 0.

当系数 a 不等于 1 时，分解因式需要格外小心。例如，解 2x² + 7x + 3 = 0。

We need two brackets with terms that give 2x² when expanded. The possible pairs are (2x …)(x …). We then test constant terms that multiply to 3: 1 and 3.

我们需要两个括号，其展开后得到 2x²。可能的组合是 (2x …)(x …)。然后测试相乘得 3 的常数项：1 和 3。

(2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3 ✓

So the equation becomes (2x + 1)(x + 3) = 0, giving x = -½ or x = -3.

因此方程变为 (2x + 1)(x + 3) = 0，得到 x = -½ 或 x = -3。

Always expand your factors to check them. One sign error can change the whole result.

始终通过展开因式来检验。一个符号错误就可能导致整个结果改变。

4. The Quadratic Formula | 二次求根公式

The quadratic formula solves any quadratic equation, even when factorisation is difficult or impossible. The formula is:

二次求根公式可以解任何二次方程，即使因式分解很困难或不可能。公式为：

x = (-b ± √(b² – 4ac)) / (2a)

To use it, substitute the values of a, b and c from ax² + bx + c = 0.

使用时，从 ax² + bx + c = 0 中代入 a、b、c 的值。

Example: solve 3x² – 4x – 2 = 0. Here a = 3, b = -4, c = -2.

例如：解 3x² – 4x – 2 = 0。这里 a = 3，b = -4，c = -2。

x = (4 ± √((-4)² – 4 × 3 × (-2))) / (2 × 3) = (4 ± √(16 + 24)) / 6 = (4 ± √40) / 6

This gives x = (4 + √40) / 6 or x = (4 – √40) / 6, which can be simplified as x = (2 ± √10) / 3.

因此 x = (4 + √40) / 6 或 x = (4 – √40) / 6，可化简为 x = (2 ± √10) / 3。

5. Completing the Square | 配方法

Completing the square rewrites x² + bx as (x + p)² + q. This method is particularly useful for finding turning points of quadratic graphs.

配方法将 x² + bx 改写为 (x + p)² + q。这种方法特别适用于求二次函数图象的顶点。

For example, solve x² + 6x + 2 = 0. Take half of 6, which is 3, and write (x + 3)² – 9 + 2 = 0.

例如，解 x² + 6x + 2 = 0。取 6 的一半为 3，写成 (x + 3)² – 9 + 2 = 0。

(x + 3)² – 7 = 0 → (x + 3)² = 7 → x + 3 = ±√7

Therefore x = -3 + √7 or x = -3 – √7.

因此 x = -3 + √7 或 x = -3 – √7。

If the coefficient of x² is not 1, first factor out the coefficient before completing the square.

如果 x² 的系数不是 1，先提出该系数，再进行配方。

6. The Discriminant and the Number of Roots | 判别式与根的数量

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

表达式 b² – 4ac 称为判别式。它告诉我们方程有多少个实数根。

If b² – 4ac > 0, there are two distinct real roots.
If b² – 4ac = 0, there is exactly one repeated root.
If b² – 4ac < 0, there are no real roots.

若 b² – 4ac > 0，方程有两个不同的实数根。
若 b² – 4ac = 0，方程只有一个重根。
若 b² – 4ac < 0，方程没有实数根。

For example, x² + x + 1 = 0 has discriminant 1² – 4 × 1 × 1 = -3, so it has no real roots.

例如，x² + x + 1 = 0 的判别式为 1² – 4 × 1 × 1 = -3，因此没有实数根。

In Edexcel IGCSE questions, you may be asked to state the nature of the roots without solving the equation.

在 Edexcel IGCSE 考题中，可能会要求你不解方程而直接说明根的性质。

7. Solving Word Problems with Quadratics | 用二次方程解应用题

Many exam problems involve a real-world situation that leads to a quadratic equation. For example, the area of a rectangle is given, and one side is expressed in terms of the other.

许多考试题目涉及实际问题，需要列出二次方程。例如，已知矩形的面积，且一边用另一边表示。

A rectangle has a length of (x + 5) cm and a width of (x – 1) cm. Its area is 24 cm². Find x.

一个矩形的长为 (x + 5) cm，宽为 (x – 1) cm，面积为 24 cm²。求 x。

(x + 5)(x – 1) = 24 → x² + 4x – 5 = 24 → x² + 4x – 29 = 0

Using the quadratic formula with a = 1, b = 4, c = -29 gives x = -2 ± √33. Since length cannot be negative, x = -2 + √33 ≈ 3.74.

使用二次求根公式，a = 1，b = 4，c = -29，得 x = -2 ± √33。由于长度不能为负，取 x = -2 + √33 ≈ 3.74。

Always check that any answer makes sense in the context of the question.

始终检查答案是否符合题目的实际意义。

8. Quadratic Graphs and the x-axis | 二次函数图象与 x 轴

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

ax² + bx + c = 0 的解，就是曲线 y = ax² + bx + c 与 x 轴交点的 x 坐标。

If the discriminant is positive, the curve crosses the x-axis twice. If it is zero, the curve touches the x-axis at one point. If it is negative, the curve does not cross the x-axis.

若判别式为正，曲线与 x 轴相交两次。若为零，曲线与 x 轴相切于一点。若为负，曲线与 x 轴不相交。

This connection between algebra and graphs is a common theme in Edexcel IGCSE questions.

代数与图象之间的联系是 Edexcel IGCSE 试题中常见的主题。

9. Solving Quadratic Equations by Drawing Graphs | 通过画图解二次方程

In some exam questions, you may be given a graph and asked to solve an equation by reading where the curve meets a line.

在某些考题中，你会得到一张图，并要求通过读取曲线与直线的交点来解方程。

For example, if y = x² – 2x – 3 is drawn, the solutions of x² – 2x – 3 = 0 are found where the curve meets the x-axis.

例如，若画出了 y = x² – 2x – 3，则 x² – 2x – 3 = 0 的解就是曲线与 x 轴的交点。

To solve x² – 2x – 5 = 0, you could draw the line y = 2 on the same axes and read off the x-values at the intersections, because x² – 2x – 5 = 0 is the same as x² – 2x – 3 = 2.

要解 x² – 2x – 5 = 0，可在同一坐标系中画直线 y = 2，然后读取交点处的 x 值，因为 x² – 2x – 5 = 0 相当于 x² – 2x – 3 = 2。

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

图象法适合估算答案，代数法给出精确值。

10. Solving Quadratic Equations with Fractions or Roots | 含分数或根号的二次方程

Sometimes a quadratic equation is not given in standard form. For example, x + 3/x = 4 can be transformed by multiplying both sides by x.

有时二次方程并非以标准形式给出。例如，x + 3/x = 4 可以通过两边乘以 x 来转换。

x + 3/x = 4 → x² + 3 = 4x → x² – 4x + 3 = 0

Factorise to get (x – 1)(x – 3) = 0, so x = 1 or x = 3. Be careful that x = 0 must be excluded before multiplying by x.

因式分解得 (x – 1)(x – 3) = 0，所以 x = 1 或 x = 3。注意在乘以 x 之前必须排除 x = 0 的情况。

Similarly, an equation like x = √(2x + 3) requires squaring both sides, then solving, and finally checking for extraneous roots.

类似地，方程如 x = √(2x + 3) 需要两边平方，然后求解，最后检查是否有增根。

11. Common Mistakes and How to Avoid Them | 常见错误及避免方法

One common mistake is forgetting to rearrange the equation into standard form before using the quadratic formula. Always write ax² + bx + c = 0 first.

一个常见错误是使用求根公式前忘记把方程整理成标准形式。务必先写成 ax² + bx + c = 0。

Another mistake is losing the negative sign when substituting b. In the formula, -b is not the same as the original b with a minus sign accidentally dropped.

另一个错误是代入 b 时丢掉负号。在公式中，-b 与原始 b 不同，不能不小心去掉负号。

Also, when factorising, do not forget that the product of two brackets equals zero means each bracket has its own solution.

此外，因式分解时，不要忘记两个括号相乘等于零意味着每个括号各自有解。

Check your solutions by substituting them back into the original equation. This takes only a few seconds and prevents careless errors.

将解代回原方程进行检验，只需几秒钟，能防止粗心错误。

12. Exam Tips for Edexcel IGCSE | Edexcel IGCSE 考试技巧

Show all your working clearly. The examiner awards method marks even if your final answer is slightly wrong.

清晰展示所有计算过程。即使最终答案略有错误，考官也会给步骤分。

If a question says “give your answer correct to 2 decimal places”, use the quadratic formula rather than factorisation, because factorisation may not give the required precision.

如果题目要求”答案精确到两位小数”，应使用求根公式而非因式分解，因为因式分解可能无法给出所需精度。

Learn the quadratic formula thoroughly; it is not given on the Edexcel IGCSE formula sheet.

彻底记住二次求根公式；Edexcel IGCSE 公式表中并不提供它。

Practice past paper questions on solving quadratic equations, including those with context, graphs, and the discriminant.

练习历年真题中关于解二次方程的题目，包括带实际情境、图象和判别式的题。

With regular practice, you will quickly recognise the best method to use in each situation.

通过定期练习，你将能迅速识别每种情境下最适合的方法。

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Solving Quadratic Equations | 解一元二次方程