Solving Quadratic Equations | 求解二次方程

📚 Solving Quadratic Equations | 求解二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, geometry, and problem-solving questions, and mastering it is essential for exam success.

二次方程是 IGCSE 数学中最重要的内容之一。它出现在代数、几何和应用题中，掌握它对于考试成功至关重要。

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

A quadratic equation is a polynomial equation of degree 2. This means the highest power of the variable, usually x, is 2. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

二次方程是次数为 2 的多项式方程。这意味着变量（通常是 x）的最高次数是 2。它的一般形式是 ax² + bx + c = 0，其中 a、b、c 是常数，且 a ≠ 0。

If a = 0, the equation becomes linear, not quadratic. Examples include x² – 5x + 6 = 0 and 2x² + 3x – 1 = 0.

如果 a = 0，方程就变成一次方程，而不是二次方程。例如 x² – 5x + 6 = 0 和 2x² + 3x – 1 = 0。

2. The Standard Form | 标准形式

Before solving any quadratic equation, it is often useful to rearrange it into the standard form ax² + bx + c = 0. This makes it easier to factorise, complete the square, or apply the quadratic formula.

在解任何二次方程之前，通常需要将其整理成标准形式 ax² + bx + c = 0。这样可以更容易地进行因式分解、配方或应用二次公式。

For example, the equation 2x² + 3 = 5x can be rewritten as 2x² – 5x + 3 = 0. Here, a = 2, b = -5, and c = 3.

例如，方程 2x² + 3 = 5x 可以改写为 2x² – 5x + 3 = 0。此时 a = 2，b = -5，c = 3。

Remember to move all terms to one side so that the right-hand side is zero.

记住要将所有项移到等号左边，使右边为 0。

3. Solving by Factorising | 因式分解法

Factorising is the fastest method when the quadratic can be written as a product of two linear expressions. For example, x² – 5x + 6 = 0 can be factorised as (x – 2)(x – 3) = 0.

当二次式可以写成两个一次因式的乘积时，因式分解法是最快的方法。例如，x² – 5x + 6 = 0 可以因式分解为 (x – 2)(x – 3) = 0。

Using the zero-product rule, if the product of two factors is zero, then at least one factor must be zero. Hence x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

根据零乘积法则，如果两个因式的乘积为零，那么至少有一个因式为零。因此 x – 2 = 0 或 x – 3 = 0，解得 x = 2 或 x = 3。

For quadratics with a ≠ 1, such as 2x² + 5x – 3 = 0, you may need to find a pair of factors that multiply to a × c = -6 and add to b = 5. The numbers 6 and -1 work, so we split the middle term:

对于 a ≠ 1 的二次方程，例如 2x² + 5x – 3 = 0，需要找到两个数，它们相乘等于 a × c = -6，相加等于 b = 5。数字 6 和 -1 满足条件，于是我们拆分中间项：

2x² + 6x – x – 3 = 0

Then factor by grouping: 2x(x + 3) – 1(x + 3) = 0, so (2x – 1)(x + 3) = 0, giving x = ½ or x = -3.

然后分组因式分解：2x(x + 3) – 1(x + 3) = 0，所以 (2x – 1)(x + 3) = 0，解得 x = ½ 或 x = -3。

4. The Quadratic Formula | 二次公式

When factorisation is difficult or impossible, the quadratic formula always works. For any quadratic equation ax² + bx + c = 0, the solutions are given by:

当因式分解困难或无法进行时，二次公式总是有效。对于任何二次方程 ax² + bx + c = 0，其解为：

x = (-b ± √(b² – 4ac)) / 2a

You should memorise this formula. It is not provided in the IGCSE formula sheet for all boards, so practise using it carefully.

你应该记住这个公式。并非所有考试局都会在公式表中提供它，所以请仔细练习使用。

Example: Solve 3x² – 4x – 2 = 0. Here a = 3, b = -4, c = -2. Substituting into the formula:

示例：解 3x² – 4x – 2 = 0。这里 a = 3，b = -4，c = -2。代入公式：

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

So x = (4 + √40) / 6 or x = (4 – √40) / 6. These can be simplified to x = (2 ± √10) / 3.

所以 x = (4 + √40) / 6 或 x = (4 – √40) / 6。可简化为 x = (2 ± √10) / 3。

5. Completing the Square | 配方法

Completing the square rewrites a quadratic in the form a(x + p)² + q. This is useful for solving equations and for finding turning points of graphs.

配方法将二次式改写为 a(x + p)² + q 的形式。这对解方程以及求图像的顶点坐标很有用。

For x² + 6x + 2 = 0, first write x² + 6x as (x + 3)² – 9. Then the equation becomes:

对于 x² + 6x + 2 = 0，先把 x² + 6x 写成 (x + 3)² – 9。于是方程变为：

(x + 3)² – 9 + 2 = 0

So (x + 3)² = 7. Taking the square root gives x + 3 = ±√7, hence x = -3 ± √7.

因此 (x + 3)² = 7。两边开平方得 x + 3 = ±√7，所以 x = -3 ± √7。

When a ≠ 1, factor a out first. For 2x² – 8x + 5 = 0, write 2(x² – 4x) + 5 = 0, then 2[(x – 2)² – 4] + 5 = 0, giving 2(x – 2)² – 3 = 0.

当 a ≠ 1 时，先提出 a。对于 2x² – 8x + 5 = 0，写成 2(x² – 4x) + 5 = 0，然后 2[(x – 2)² – 4] + 5 = 0，得到 2(x – 2)² – 3 = 0。

6. The Discriminant | 判别式

The discriminant is the expression Δ = b² – 4ac inside the square root of the quadratic formula. It tells us the number and nature of the roots without solving the equation.

判别式是二次公式根号内的表达式 Δ = b² – 4ac。它可以在不解方程的情况下告诉我们根的数量和性质。

If Δ > 0, there are two distinct real roots.
如果 Δ > 0，方程有两个不相等的实数根。
If Δ = 0, there is exactly one repeated real root.
如果 Δ = 0，方程有一个重根（两个相等的实数根）。
If Δ < 0, there are no real roots (two complex roots).
如果 Δ < 0，方程没有实数根（有两个复数根）。

For example, for x² – 4x + 4 = 0, Δ = 16 – 16 = 0, so there is one repeated root: x = 2.

例如，对于 x² – 4x + 4 = 0，Δ = 16 – 16 = 0，所以有一个重根：x = 2。

Questions often ask you to find the range of k for which x² + 2kx + 4 = 0 has real roots. Set Δ ≥ 0:

题目常要求你求 x² + 2kx + 4 = 0 有实数根时 k 的取值范围。令 Δ ≥ 0：

(2k)² – 4×1×4 ≥ 0 → 4k² – 16 ≥ 0 → k² ≥ 4

So k ≥ 2 or k ≤ -2.

所以 k ≥ 2 或 k ≤ -2。

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 and product of the roots are related to the coefficients:

对于二次方程 ax² + bx + c = 0，若其根为 α 和 β，则根的和与积与系数有如下关系：

α + β = -b/a, αβ = c/a

This relation is often used to find the value of expressions such as α² + β² without solving for α and β.

这个关系常用于在不解出 α 和 β 的情况下求 α² + β² 等表达式的值。

For example, if 2x² + 6x – 5 = 0 has roots α and β, then α + β = -6/2 = -3 and αβ = -5/2.

例如，如果 2x² + 6x – 5 = 0 的根为 α 和 β，则 α + β = -6/2 = -3，αβ = -5/2。

To find α² + β², use the identity:

要求 α² + β²，使用恒等式：

α² + β² = (α + β)² – 2αβ = (-3)² – 2(-5/2) = 9 + 5 = 14

8. Applications | 实际应用

Quadratic equations often appear in word problems involving areas, projectile motion, and number relationships. For example, the area of a rectangle is 30 cm² and its length is 7 cm longer than its width. Find the width.

二次方程经常出现在涉及面积、抛体运动和数字关系的应用题中。例如，一个长方形的面积为 30 cm²，其长比宽多 7 cm，求宽。

Let the width be x cm. Then the length is x + 7 cm. The area equation is x(x + 7) = 30, which expands to x² + 7x – 30 = 0.

设宽为 x cm，则长为 x + 7 cm。面积方程为 x(x + 7) = 30，展开得 x² + 7x – 30 = 0。

Factorising gives (x + 10)(x – 3) = 0, so x = -10 or x = 3. Since width cannot be negative, x = 3 cm.

因式分解得 (x + 10)(x – 3) = 0，所以 x = -10 或 x = 3。因为宽度不能为负数，所以 x = 3 cm。

Always check whether your answer makes sense in the context of the problem.

始终检查你的答案在问题情境中是否有意义。

9. Common Mistakes | 常见错误

Students often make avoidable errors when solving quadratic equations. Here are the most common ones to watch out for:

学生在解二次方程时常常犯一些可以避免的错误。以下是最常见的几种，需要特别注意：

Forgetting to rearrange into standard form before factorising or applying the formula.
忘记先将方程整理成标准形式，再进行因式分解或应用公式。
Diving both sides by x when x = 0 is a possible solution, which loses roots.
在 x = 0 可能是解时，两边同时除以 x，从而丢失根。
Incorrectly writing the quadratic formula, especially confusing the sign of -b.
写错二次公式，特别是弄错 -b 的符号。
Making arithmetic mistakes inside the square root of the discriminant.
在判别式的根号内出现算术错误。
Forgetting that √(x²) = |x|, not simply x, when solving by square roots.
用开平方解方程时，忘记 √(x²) = |x|，而不是简单地等于 x。

Always test your solutions by substituting them back into the original equation.

始终将解代回原方程进行检验。

10. Practice Tips | 练习建议

To succeed with quadratic equations, practise the three main methods until they become automatic. Start with simple factorisable examples, then move to the quadratic formula and completing the square.

要在二次方程上取得好成绩，请练习三种主要方法直到熟练掌握。从简单的可因式分解的示例开始，然后转向二次公式和配方法。

Use past paper questions to familiarise yourself with the different styles of questions, including those with the discriminant and word problems. Remember to show clear steps in your working, as method marks are often awarded even when the final answer is wrong.

使用历年真题来熟悉不同类型的问题，包括涉及判别式和实际应用的问题。记住在解答中展示清晰的步骤，因为即使最终答案错误，方法分常常也会被授予。

Finally, review your mistakes carefully and avoid repeating them. Consistency is the key to mastering this topic.

最后，仔细复习你的错误，避免重复。持续练习是掌握这一主题的关键。

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Solving Quadratic Equations | 求解二次方程