Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. In IGCSE mathematics, solving these equations is a core skill that appears in both non-calculator and calculator papers.

二次方程是指最高次数为二次的方程，它至少包含一个平方项。在 IGCSE 数学中，解这类方程是一项核心技能，既出现在非计算器试卷中，也出现在允许使用计算器的试卷中。

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 constants and a ≠ 0. The variable x represents an unknown number.

二次方程是可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 是常数，且 a ≠ 0。变量 x 代表一个未知数。

For example, 2x² + 3x – 5 = 0 is a quadratic equation, while 3x + 2 = 0 is linear because it has no x² term.

例如，2x² + 3x – 5 = 0 是一个二次方程，而 3x + 2 = 0 是一次方程，因为它没有 x² 项。

2. Standard Form | 标准形式

To solve a quadratic equation easily, it is usually helpful to rearrange it into the standard form ax² + bx + c = 0. This means moving all terms to one side, leaving zero on the other.

为了更容易地解二次方程，通常需要把它整理成标准形式 ax² + bx + c = 0。也就是说，把所有项移到一边，另一边为 0。

For instance, x² = 5x – 6 is not in standard form. Subtract 5x and add 6 to both sides to get x² – 5x + 6 = 0.

例如，x² = 5x – 6 不是标准形式。两边同时减去 5x 并加上 6，得到 x² – 5x + 6 = 0。

If a ≠ 0, then ax² + bx + c = 0 is the standard quadratic form.

如果 a ≠ 0，那么 ax² + bx + c = 0 就是二次方程的标准形式。

3. Solving by Factorisation | 因式分解法

Factorisation is often the quickest method when the quadratic has simple integer factors. The idea is to write the expression as a product of two brackets, then set each bracket equal to zero.

当二次式具有简单的整数因式时，因式分解往往是最快的方法。思路是把表达式写成两个括号的乘积，然后令每个括号等于零。

Example: Solve x² – 5x + 6 = 0.

例：解方程 x² – 5x + 6 = 0。

(x – 2)(x – 3) = 0

So x – 2 = 0 or x – 3 = 0, giving x = 2 or x = 3.

所以 x – 2 = 0 或 x – 3 = 0，得到 x = 2 或 x = 3。

For quadratics with a leading coefficient a ≠ 1, you may need to factor by grouping or use trial and error.

当二次项系数 a ≠ 1 时，可能需要使用分组分解法或尝试法。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² = q, which then allows you to solve by taking square roots.

配方法将二次方程改写为 (x + p)² = q 的形式，然后通过取平方根来求解。

Example: Solve x² + 6x + 1 = 0 by completing the square.

例：用配方法解 x² + 6x + 1 = 0。

x² + 6x = -1

(x + 3)² – 9 = -1

(x + 3)² = 8

Then x + 3 = ±√8, so x = -3 ± 2√2.

因此 x + 3 = ±√8，所以 x = -3 ± 2√2。

This method is especially useful when the quadratic cannot be factorised easily.

当二次方程不容易因式分解时，这种方法尤其有用。

5. The Quadratic Formula | 二次公式

For any quadratic equation ax² + bx + c = 0, the solutions can be found using the quadratic formula. This formula works for all quadratics, including those with irrational or complex roots.

对于任意二次方程 ax² + bx + c = 0，都可以使用二次公式求根。该公式适用于所有二次方程，包括具有无理根或复数根的情况。

x = (-b ± √(b² – 4ac)) / (2a)

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

例：用公式解 2x² + 3x – 2 = 0。

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

这里 a = 2，b = 3，c = -2。代入公式：

x = (-3 ± √(3² – 4·2·(-2))) / (2·2)

x = (-3 ± √(9 + 16)) / 4

x = (-3 ± 5) / 4

So x = 0.5 or x = -2.

所以 x = 0.5 或 x = -2。

6. The Discriminant | 判别式

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

根号内的表达式 b² – 4ac 称为判别式。它告诉我们二次方程有多少个实数根。

If b² – 4ac > 0, there are two distinct real roots.
如果 b² – 4ac > 0，则方程有两个不相等的实数根。
If b² – 4ac = 0, there is exactly one real root (a repeated root).
如果 b² – 4ac = 0，则方程有一个实数根（重根）。
If b² – 4ac < 0, there are no real roots (two complex roots).
如果 b² – 4ac < 0，则方程没有实数根（有两个复数根）。

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

例如，对于 x² – 4x + 4 = 0，判别式为 16 – 16 = 0，所以有一个重根 x = 2。

7. Solving Quadratic Equations by Graphing | 图像法

The solutions of a quadratic equation ax² + bx + c = 0 are the x-coordinates of the points where the graph of y = ax² + bx + c crosses the x-axis.

二次方程 ax² + bx + c = 0 的解，就是抛物线 y = ax² + bx + c 与 x 轴交点的横坐标。

If the graph touches the x-axis at exactly one point, the equation has one repeated root. If it does not touch the x-axis, there are no real roots.

如果图像与 x 轴只有一个交点，则方程有一个重根。如果不与 x 轴相交，则没有实数根。

Graphing provides a visual understanding, but in exams algebraic methods are usually required for exact answers.

图像法提供了直观的理解，但在考试中通常需要用代数方法求出精确答案。

8. Word Problems | 应用题

Many word problems involve quadratic equations. To solve them, define a variable, form a quadratic equation from the given information, solve it, and then check whether the answers are reasonable in the context.

许多应用题涉及二次方程。解法步骤包括：设定变量，根据题意建立二次方程，求解，然后检验答案是否符合实际情境。

Example: The area of a rectangle is 20 cm² and its length is 3 cm more than its width. Find the width.

例：一个矩形的面积为 20 平方厘米，长比宽多 3 厘米。求宽。

Let w be the width. Then length = w + 3. The area is w(w + 3) = 20, so w² + 3w – 20 = 0. Using the formula gives w = (-3 ± √(9 + 80)) / 2 = (-3 ± √89) / 2. The positive solution is approximately 3.22 cm.

设宽为 w，则长为 w + 3。面积为 w(w + 3) = 20，即 w² + 3w – 20 = 0。使用公式得 w = (-3 ± √(9 + 80)) / 2 = (-3 ± √89) / 2。正数解约为 3.22 厘米。

9. Common Mistakes | 常见错误

Forgetting to rearrange the equation into standard form before factorising or using the formula.
忘记在使用因式分解或公式前把方程整理成标准形式。
Losing a negative sign when substituting into the quadratic formula.
代入二次公式时丢掉负号。
Thinking that x² = 4 gives only x = 2, when it actually gives x = ±2.
认为 x² = 4 只有 x = 2，而实际上 x = ±2。
Dividing both sides of an equation by a variable, which may cause loss of a root.
方程两边除以同一个变量，这可能会丢失一个根。

To avoid these errors, always check your solutions by substituting back into the original equation.

为避免这些错误，总是将解代回原方程进行检验。

10. Practice Questions | 练习题

Try solving these quadratic equations using any method:

尝试用任意方法解下列二次方程：

1. x² – 7x + 10 = 0	2. 3x² + 5x – 2 = 0
3. x² + 4x – 7 = 0	4. 2x² – 8x + 8 = 0

Answers: 1. x = 2 or 5; 2. x = 1/3 or -2; 3. x = -2 ± √11; 4. x = 2 (repeated).

答案：1. x = 2 或 5；2. x = 1/3 或 -2；3. x = -2 ± √11；4. x = 2（重根）。

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Solving Quadratic Equations | 解二次方程