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

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。

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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。

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。

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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)

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。

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

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。

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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

mple, 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。

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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

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