IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解一元二次方程

📚 IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解一元二次方程

A quadratic equation is one of the most common question types in IGCSE Mathematics. You will be expected to solve quadratics by factorising, completing the square, and using the quadratic formula. You will also need to interpret the discriminant and apply quadratic equations to real-life problems.

二次方程是 IGCSE 数学中最常见的题型之一。你需要掌握因式分解法、配方法和公式法来解二次方程，还需要理解判别式的意义，并能够把二次方程应用到实际生活问题中。

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

A quadratic equation is an equation where the highest power of the unknown is 2. Its general form is usually written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. If a were zero, the equation would become linear, not quadratic.

二次方程是指未知数的最高次数为 2 的方程。它的一般形式通常写作 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。如果 a 等于 0，该方程就会变成一次方程，而不是二次方程。

For example, x² + 5x + 6 = 0 and 2x² − 3x + 1 = 0 are both quadratic equations. The expression x² + 5x + 6 is called a quadratic expression, while the whole statement with an equals sign is a quadratic equation.

例如，x² + 5x + 6 = 0 和 2x² − 3x + 1 = 0 都是二次方程。表达式 x² + 5x + 6 叫做二次式，而带有等号的整个式子就是二次方程。

2. Standard Form and Key Terminology | 标准形式与关键术语

Before solving a quadratic equation, you should rearrange it into standard form ax² + bx + c = 0. This means putting all terms on one side of the equation and setting the other side equal to zero.

在解二次方程之前，你应该先把它整理成标准形式 ax² + bx + c = 0。这意味着把所有项移到方程的一边，并让另一边等于零。

The coefficient a is the number multiplying x², b is the number multiplying x, and c is the constant term. For example, in 3x² − 7x + 2 = 0, we have a = 3, b = −7 and c = 2.

系数 a 是乘以 x² 的数，b 是乘以 x 的数，c 是常数项。例如，在 3x² − 7x + 2 = 0 中，a = 3，b = −7，c = 2。

Being careful with negative coefficients is important. If b is negative, it must be written as a negative value in the formula or when factorising.

处理负系数时要特别小心。如果 b 是负数，在公式法或因式分解中都必须把它写作负值。

3. Solving by Factorising: The Zero Product Principle | 因式分解法与零乘积原理

Factorising is often the fastest method when the quadratic expression can be written as a product of two linear brackets. The key idea is the zero product principle: if two factors multiply to give zero, then at least one of the factors must be zero.

当二次式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。核心思想是零乘积原理：如果两个因式相乘等于零，那么至少有一个因式必须为零。

This means if (x + p)(x + q) = 0, then either x + p = 0 or x + q = 0. Solving these two simple linear equations gives the two possible solutions of the original quadratic equation.

这意味着如果 (x + p)(x + q) = 0，那么要么 x + p = 0，要么 x + q = 0。解这两个简单的一次方程，就可以得到原二次方程的两个可能解。

Solve x + p = 0 to get x = −p.
Solve x + q = 0 to get x = −q.

解 x + p = 0，得到 x = −p。
解 x + q = 0，得到 x = −q。

Always check your factorisation by expanding the brackets to make sure it matches the original quadratic expression.

一定要通过展开括号来检验因式分解是否正确，确保它与原来的二次式一致。

4. Worked Example: Factorising x² + bx + c | 例题：分解 x² + bx + c 型

Consider the equation x² + 5x + 6 = 0. We need two numbers that multiply to give c = 6 and add to give b = 5. The numbers 2 and 3 satisfy both conditions because 2 × 3 = 6 and 2 + 3 = 5.

考虑方程 x² + 5x + 6 = 0。我们需要找到两个数，它们相乘等于 c = 6，相加等于 b = 5。数字 2 和 3 满足这两个条件，因为 2 × 3 = 6，且 2 + 3 = 5。

So the equation can be written as (x + 2)(x + 3) = 0. This gives x + 2 = 0 or x + 3 = 0, so the solutions are x = −2 and x = −3.

因此方程可以写成 (x + 2)(x + 3) = 0。于是得到 x + 2 = 0 或 x + 3 = 0，所以解为 x = −2 和 x = −3。

If the constant term is positive but the middle coefficient is negative, both numbers must be negative. For example, x² − 7x + 12 = 0 factorises to (x − 3)(x − 4) = 0 because −3 and −4 multiply to give 12 and add to give −7.

如果常数项为正但中间系数为负，那么这两个数都必须为负。例如，x² − 7x + 12 = 0 可以因式分解为 (x − 3)(x − 4) = 0，因为 −3 和 −4 相乘等于 12，相加等于 −7。

5. Worked Example: Factorising ax² + bx + c | 例题：分解 ax² + bx + c 型

When the leading coefficient a is not 1, factorising requires more thought. Consider 2x² + 7x + 3 = 0. We need two numbers that multiply to give 2 × 3 = 6 and add to give 7. The numbers are 6 and 1.

当首项系数 a 不为 1 时，因式分解需要更仔细。考虑 2x² + 7x + 3 = 0。我们需要找到两个数，它们相乘等于 2 × 3 = 6，相加等于 7。这两个数是 6 和 1。

Rewrite the middle term using these numbers: 2x² + 6x + x + 3 = 0. Then factorise in pairs: 2x(x + 3) + 1(x + 3) = 0. This gives (2x + 1)(x + 3) = 0.

用这两个数改写中间项：2x² + 6x + x + 3 = 0。然后分组分解：2x(x + 3) + 1(x + 3) = 0。于是得到 (2x + 1)(x + 3) = 0。

Solving 2x + 1 = 0 gives x = −1/2, and solving x + 3 = 0 gives x = −3. So the solutions are x = −1/2 and x = −3.

解 2x + 1 = 0 得到 x = −1/2，解 x + 3 = 0 得到 x = −3。因此解为 x = −1/2 和 x = −3。

6. Completing the Square | 配方法

Completing the square is a useful method when a quadratic cannot easily be factorised. It also helps when you need to find the vertex of a parabola or solve equations involving irrational roots.

配方法适用于二次方程不容易因式分解的情况。当你需要求抛物线的顶点，或者解含有无理根的方程时，配方法也非常有用。

For an equation in the form x² + bx + c = 0, you can rewrite x² + bx as (x + b/2)² − (b/2)². This process is called completing the square.

对于形如 x² + bx + c = 0 的方程，你可以把 x² + bx 改写为 (x + b/2)² − (b/2)²。这个过程叫做配方。

For example, solve x² + 6x + 2 = 0. First, rewrite it as (x + 3)² − 9 + 2 = 0, which simplifies to (x + 3)² − 7 = 0. Therefore (x + 3)² = 7, so x + 3 = ±√7, giving x = −3 ± √7.

例如，解 x² + 6x + 2 = 0。首先把它改写为 (x + 3)² − 9 + 2 = 0，化简得 (x + 3)² − 7 = 0。因此 (x + 3)² = 7，所以 x + 3 = ±√7，得到 x = −3 ± √7。

x² + bx = (x + b/2)² − (b/2)²

7. The Quadratic Formula | 二次公式

The quadratic formula can solve any quadratic equation in standard form ax² + bx + c = 0. It is especially useful when factorising is difficult or when the coefficients are large or non-integer.

二次公式可以解任何标准形式 ax² + bx + c = 0 的二次方程。当因式分解困难，或者系数较大、不是整数时，二次公式尤其有用。

x = [−b ± √(b² − 4ac)] / (2a)

To use the formula, first identify a, b and c. For example, in 2x² − 4x − 6 = 0, a = 2, b = −4 and c = −6. Substituting gives x = [4 ± √(16 + 48)] / 4, which simplifies to x = [4 ± √64] / 4 = [4 ± 8] / 4.

使用公式时，首先要找出 a、b、c。例如，在 2x² − 4x − 6 = 0 中，a = 2，b = −4，c = −6。代入公式得到 x = [4 ± √(16 + 48)] / 4，化简为 x = [4 ± √64] / 4 = [4 ± 8] / 4。

This gives x = (4 + 8) / 4 = 3 or x = (4 − 8) / 4 = −1. Always write the formula clearly and substitute carefully to avoid sign errors.

于是得到 x = (4 + 8) / 4 = 3 或 x = (4 − 8) / 4 = −1。一定要清楚写出公式并仔细代入，避免符号错误。

8. Discriminant and Number of Roots | 判别式与根的个数

The discriminant is the part of the quadratic formula under the square root: Δ = b² − 4ac. It tells you how many real solutions a quadratic equation has before you solve it.

判别式是二次公式中根号下面的部分：Δ = b² − 4ac。它可以让你在求解之前判断二次方程有多少个实数解。

Condition	Number of real roots
Δ > 0	Two distinct real roots
Δ = 0	One repeated real root
Δ < 0	No real roots

条件	实数根的个数
Δ > 0	两个不同的实数根
Δ = 0	一个重复的实数根
Δ < 0	没有实数根

For example, for x² + 2x + 3 = 0, the discriminant is 2² − 4 × 1 × 3 = 4 − 12 = −8, which is negative, so the equation has no real roots.

例如，对于 x² + 2x + 3 = 0，判别式为 2² − 4 × 1 × 3 = 4 − 12 = −8，是负数，所以这个方程没有实数根。

9. Quadratic Word Problems: Area and Motion | 二次方程应用题：面积与运动

IGCSE questions often hide quadratic equations inside word problems. A common type involves the area of a rectangle where one side is expressed in terms of the other, leading to an equation such as x(x + 3) = 40.

IGCSE 考试常把二次方程隐藏在实际应用题中。常见的一种类型涉及矩形面积，其中一条边用另一条边表示，从而得到形如 x(x + 3) = 40 的方程。

For example, if a rectangle has width x and length x + 3, and its area is 40 cm², then x(x + 3) = 40. Expanding gives x² + 3x − 40 = 0. Factorising gives (x + 8)(x − 5) = 0, so x = 5 or x = −8. Since length cannot be negative, x = 5 cm.

例如，如果一个矩形的宽为 x，长为 x + 3，面积为 40 cm²，那么 x(x + 3) = 40。展开得 x² + 3x − 40 = 0。因式分解得 (x + 8)(x − 5) = 0，所以 x = 5 或 x = −8。由于长度不能为负，所以 x = 5 cm。

Another common context is vertical motion under gravity, where the height of an object can be modelled by a quadratic equation. Solving the equation tells you when the object reaches a given height.

另一种常见情境是重力作用下的竖直运动，物体高度可以用二次方程建模。解这个方程可以告诉你物体何时到达某一给定高度。

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

One common mistake is forgetting to rearrange the equation into the form ax² + bx + c = 0 before factorising or using the formula. If you try to solve x² + 5x = 6 by factorising the left side immediately, you miss the need to subtract 6 from both sides first.

一个常见错误是在因式分解或使用公式前，忘记把方程整理成 ax² + bx + c = 0 的形式。如果你直接对 x² + 5x = 6 的左边进行因式分解，就会忽略首先要把 6 从两边减掉。

Another common error is mishandling negative signs in the quadratic formula, especially when b is negative. Always use brackets when substituting: for b = −4, write −(−4) = 4 clearly.

另一个常见错误是在二次公式中处理负号不当，尤其是当 b 为负数时。代入时一定要使用括号：例如 b = −4，要清楚写成 −(−4) = 4。

Finally, always check whether your answers make sense in the context of the problem. If a length or time is negative, it should usually be rejected. Write your final solutions clearly and check by substituting them back into the original equation.

最后，一定要检查你的答案在题目情境中是否合理。如果长度或时间为负数，通常应该舍去。清楚写出最终解，并代入原方程检验。

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IGCSE Maths: Solving Quadratic Equations | IGCSE 数学：解一元二次方程