Solving Quadratic Equations with the Quadratic Formula: Methods and Examples | 用二次求根公式解方程：方法与例题

📚 Solving Quadratic Equations with the Quadratic Formula: Methods and Examples | 用二次求根公式解方程：方法与例题

Quadratic equations appear regularly in Edexcel IGCSE Mathematics. While factorisation is often quick, not every quadratic factorises with integer values. The quadratic formula solves all quadratics of the form ax² + bx + c = 0 and is especially useful when exact surd answers are needed. This guide explains the method, the role of the discriminant, and gives exam-style worked examples.

二次方程在 Edexcel IGCSE 数学中经常出现。因式分解通常很快，但并非所有二次方程都能用整数因式分解。二次求根公式可以解所有形如 ax² + bx + c = 0 的二次方程，在需要给出根号形式的精确答案时尤其有用。本指南将讲解方法、判别式的作用，并给出考试风格的例题。

---

1. Standard Form | 标准形式

Before using the formula, a quadratic equation must be written in the standard form:

使用求根公式之前，必须先把二次方程写成标准形式：

ax² + bx + c = 0

Here a, b and c are real numbers, and a must not be zero. If the equation contains brackets, expand and simplify. If terms appear on both sides, move them all to one side.

其中 a、b、c 是实数，且 a 不能为零。如果方程含有括号，先展开并化简；如果等号两边都有项，则把所有项移到同一边。

For example, 2x² + 5x = 3 becomes 2x² + 5x − 3 = 0. Similarly, 4x − 3x² = 7 becomes −3x² + 4x − 7 = 0, which can be multiplied by −1 to give 3x² − 4x + 7 = 0.

例如，2x² + 5x = 3 应改写为 2x² + 5x − 3 = 0。同样，4x − 3x² = 7 应改写为 −3x² + 4x − 7 = 0，也可以两边乘以 −1，得到 3x² − 4x + 7 = 0。

---

2. The Quadratic Formula | 二次求根公式

For ax² + bx + c = 0, the solutions are given by:

对于 ax² + bx + c = 0，解由以下公式给出：

x = (-b ± √(b² − 4ac)) / (2a)

The ± sign means that there are two possible answers: one using the plus sign and one using the minus sign.

± 号表示有两个可能的答案：一个用加号，一个用减号。

To use the formula, substitute the values of a, b and c, evaluate the expression carefully, and then simplify the result.

使用该公式时，把 a、b、c 的值代入，仔细计算表达式，然后化简结果。

---

3. How the Formula Is Obtained | 公式的由来

The quadratic formula comes from completing the square on ax² + bx + c = 0. Since a ≠ 0, divide every term by a:

二次求根公式通过对 ax² + bx + c = 0 配方得到。因为 a ≠ 0，先两边除以 a：

x² + (b/a)x + c/a = 0

Then complete the square:

然后配方：

(x + b/(2a))² = (b² − 4ac)/(4a²)

Taking square roots and rearranging gives the formula. This is why the discriminant b² − 4ac appears inside the square root.

两边开平方并整理，就得到求根公式。这就是判别式 b² − 4ac 出现在根号内的原因。

---

4. Identifying a, b and c | 确定 a、b、c

After writing the equation in standard form, read a, b and c from the corresponding terms. Pay close attention to negative signs.

将方程写成标准形式后，从对应项中读出 a、b、c。要特别留意负号。

Equation	a	b	c
3x² − 7x + 2 = 0	3	−7	2
x² + 4x = 0	1	4	0
x² − 16 = 0	1	0	−16

If b or c is zero, the equation is still quadratic. For example, x² − 16 = 0 can be solved by factorising, but the quadratic formula would also work.

如果 b 或 c 为零，方程仍然是二次方程。例如，x² − 16 = 0 可以用因式分解求解，但使用求根公式也同样可行。

---

5. The Discriminant | 判别式

The expression b² − 4ac is called the discriminant. It tells us the nature of the roots without fully solving the equation.

式子 b² − 4ac

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）