Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is one of the most important topics in IGCSE Mathematics. It appears in algebra, graphs, geometry and even real-life problems. This article explains how to solve quadratic equations step by step, using factorisation, completing the square, the quadratic formula and graphical methods.

二次方程是 IGCSE 数学中最重要的话题之一。它出现在代数、图像、几何甚至实际生活问题中。本文将逐步讲解如何用因式分解、配方法、二次公式和图像法求解二次方程。

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 variable is 2. The general form is \(ax^2+bx+c=0\), but we are not using LaTeX here. So we write it as: ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是变量最高次数为 2 的方程。一般形式为 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 of quadratic equations are: 2x² + 3x – 5 = 0, x² – 4 = 0, and 3x² = 6x + 1.

如果 a = 0，方程就变成一次方程，而不是二次方程。二次方程的例子有：2x² + 3x – 5 = 0，x² – 4 = 0 以及 3x² = 6x + 1。

2. The Standard Form | 标准形式

Before solving a quadratic equation, you should always rearrange it into the standard form: ax² + bx + c = 0. This means all terms are on one side and zero is on the other side.

在解二次方程之前，你应当先将它整理为标准形式：ax² + bx + c = 0。也就是说，所有项都在一边，另一边为 0。

For example, x² = 5x – 6 must be rearranged to x² – 5x + 6 = 0. Only then can you apply the solving methods.

例如，x² = 5x – 6 必须整理成 x² – 5x + 6 = 0。只有这样才能使用各种解法。

x² – 5x + 6 = 0

3. Solving by Factorising | 因式分解法

Factorising is often the quickest method. If you can write the quadratic as a product of two linear factors, then you can find the roots by setting each factor equal to zero.

因式分解通常是最快的方法。如果你能把二次式写成两个一次因式的乘积，那么就可以令每个因式等于零来求出根。

For example, solve x² – 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give -5. These numbers are -2 and -3. So:

例如，解 x² – 5x + 6 = 0。我们要找两个数，它们相乘得 6，相加得 -5。这两个数是 -2 和 -3。于是：

(x – 2)(x – 3) = 0

Therefore x – 2 = 0 or x – 3 = 0, which gives x = 2 or x = 3.

因此 x – 2 = 0 或 x – 3 = 0，解得 x = 2 或 x = 3。

For equations like x² – 9 = 0, use the difference of two squares: (x – 3)(x + 3) = 0, so x = ±3.

对于像 x² – 9 = 0 的方程，可用平方差公式：(x – 3)(x + 3) = 0，所以 x = ±3。

4. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² + q = 0 or (x – p)² + q = 0. This method is especially useful when the equation cannot be factorised easily.

配方法把二次式改写成 (x + p)² + q = 0 或 (x – p)² + q = 0 的形式。当方程不容易因式分解时，这个方法特别有用。

Example: x² + 6x + 2 = 0. First, take half of 6, which is 3, and write (x + 3)² – 9 + 2 = 0.

例：x² + 6x + 2 = 0。首先取 6 的一半，即 3，写成 (x + 3)² – 9 + 2 = 0。

(x + 3)² – 7 = 0

Then (x + 3)² = 7, so x + 3 = ±√7, and finally x = -3 ± √7.

然后 (x + 3)² = 7，所以 x + 3 = ±√7，最终 x = -3 ± √7。

If the coefficient of x² is not 1, first divide the whole equation by that coefficient.

如果 x² 的系数不是 1，先给整个方程除以该系数。

5. The Quadratic Formula | 二次公式

The quadratic formula works for every quadratic equation, including those that cannot be factorised. The formula is given below.

二次公式适用于所有二次方程，包括那些无法因式分解的方程。公式如下。

x = (-b ± √(b² – 4ac)) / (2a)

To use it, identify a, b and c from the equation ax² + bx + c = 0. Then substitute them into the formula.

使用时，从方程 ax² + bx + c = 0 中确定 a、b、c，然后把它们代入公式。

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

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

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4

So x = (4 ± 2√10) / 4 = (2 ± √10) / 2. Give your answer to a suitable degree of accuracy if required.

所以 x = (4 ± 2√10) / 4 = (2 ± √10) / 2。若题目要求，可按合适的精度给出答案。

6. The Discriminant | 判别式

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

根号内的部分 b² – 4ac 叫做判别式。它告诉我们二次方程有多少个实数根。

If b² – 4ac > 0, the equation has two distinct real roots.

如果 b² – 4ac > 0，方程有两个不相等的实数根。
If b² – 4ac = 0, the equation has exactly one real root (a repeated root).

如果 b² – 4ac = 0，方程恰好有一个实数根（重根）。
If b² – 4ac < 0, the equation has no real roots (only complex roots).

如果 b² – 4ac < 0，方程没有实数根（只有复数根）。

For IGCSE, you may be asked to find the value of a parameter so that the equation has equal roots, using the condition b² – 4ac = 0.

在 IGCSE 中，你可能会被要求利用 b² – 4ac = 0 的条件求某个参数的值，使方程有等根。

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，就是函数 y = ax² + bx + c 的图像与 x 轴交点的 x 坐标。

To solve by graphing, sketch the parabola. The roots are the x-intercepts. If the graph does not touch the x-axis, there are no real solutions.

用图像法求解时，画出抛物线草图。根就是与 x 轴的交点。如果图像不与 x 轴相交，就没有实数解。

For example, y = x² – x – 2 crosses the x-axis at x = -1 and x = 2, so the solutions are x = -1 and x = 2.

例如，y = x² – x – 2 与 x 轴交于 x = -1 和 x = 2，所以解为 x = -1 和 x = 2。

Graphing is also useful for estimating roots, especially when exact values are not easy to calculate.

图像法也适用于估算根，尤其在精确值不易计算的时候。

8. Word Problems | 应用题

Many real-life problems lead to quadratic equations. For example, the area of a rectangle is given by length × width. If length is 5 cm more than width and area is 24 cm², we can write:

许多实际问题会导出二次方程。例如，矩形面积等于长 × 宽。若长比宽多 5 cm，面积为 24 cm²，我们可以写出：

x(x + 5) = 24

Expanding gives x² + 5x – 24 = 0, then factorise to get (x + 8)(x – 3) = 0, so x = -8 or x = 3. Since width cannot be negative, the width is 3 cm.

展开得 x² + 5x – 24 = 0，因式分解得 (x + 8)(x – 3) = 0，所以 x = -8 或 x = 3。因为宽不能为负数，所以宽为 3 cm。

Always check whether each solution makes sense in the original context. Discard negative lengths, times or other impossible values.

始终检查每个解在原始情境中是否有意义。舍弃负长度、负时间等不可能的值。

9. Common Mistakes | 常见错误

Many students lose marks on quadratic equations because of small errors. Here are some common mistakes to avoid.

许多学生在二次方程上因为小错误丢分。下面是一些要避免的常见错误。

Forgetting to rearrange the equation into standard form before factorising or using the formula.

在因式分解或使用公式之前，忘记把方程整理成标准形式。
When using the quadratic formula, forgetting that -b changes the sign of b. For example, if b = -4, then -b = 4.

使用二次公式时，忘记 -b 会改变 b 的符号。例如，若 b = -4，则 -b = 4。
Forgetting to divide by 2a when a is not 1.

当 a 不是 1 时，忘记除以 2a。
Mismatching the factors: x² + 5x + 6 = (x + 2)(x + 3), not (x + 2)(x – 3).

因式分解错误：x² + 5x + 6 = (x + 2)(x + 3)，不是 (x + 2)(x – 3)。
Throwing away negative roots without checking whether they are valid in the problem.

不检查负根在问题中是否有效就直接舍去。

10. Practice Questions | 练习题

Try these questions on your own. Answers are given below.

请独立尝试以下问题。答案见下方。

Solve x² – 7x + 12 = 0.

解 x² – 7x + 12 = 0。
Solve 2x² + 3x – 2 = 0 using the quadratic formula.

用二次公式解 2x² + 3x – 2 = 0。
Find the value of k if the equation x² + kx + 9 = 0 has a repeated root.

若方程 x² + kx + 9 = 0 有重根，求 k 的值。
A right-angled triangle has legs x and x + 2, and hypotenuse 10. Find x.

一个直角三角形两条直角边为 x 和 x + 2，斜边为 10。求 x。

Answers: 1. x = 3 or 4; 2. x = 0.5 or -2; 3. k = ±6; 4. x = 6 (since x = -8 is impossible).

答案：1. x = 3 或 4；2. x = 0.5 或 -2；3. k = ±6；4. x = 6（因为 x = -8 不可能）。

11. Summary | 总结

A quadratic equation has the form ax² + bx + c = 0. You can solve it by factorising, completing the square, using the quadratic formula, or graphing. The discriminant b² – 4ac tells you how many real roots exist.

二次方程具有形式 ax² + bx + c = 0。你可以通过因式分解、配方法、二次公式或图像法来求解。判别式 b² – 4ac 告诉你存在多少个实数根。

Always write the equation in standard form first, and always check your answers by substituting them back into the original equation. With regular practice, you will avoid common mistakes and solve these problems quickly and accurately.

先把方程写成标准形式，并且始终通过代回原方程来检查答案。通过经常练习，你将避免常见错误，并快速准确地解决这些问题。

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Solving Quadratic Equations | 解二次方程