Solving Quadratic Equations | 二次方程的解法

📚 Solving Quadratic Equations | 二次方程的解法

nstants and a ≠ 0. Being able to solve quadratics quickly and accurately is essential for both Core and Extended papers. This guide covers factorisation, completing the square, the quadratic formula, the discriminant and exam strategies.

二次方程贯穿 IGCSE 数学课程，从代数运算到现实问题建模都会出现。二次方程是指可以写成 ax² + bx + c = 0 形式的方程，其中 a、b 和 c 是常数且 a ≠ 0。无论 Core 还是 Extended 试卷，快速且准确地解二次方程都是关键能力。本指南涵盖因式分解法、配方法、二次公式法、判别式以及考试策略。

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

quadratic equations. If the coefficient of x² is zero, the equation is no longer quadratic; it becomes linear.

二次方程是二次多项式方程，变量 x 的最高次数是 2。例如 x² − 5x + 6 = 0 和 2x² + 3x − 4 = 0 都是二次方程。如果 x² 的系数为零，方程就不再是二次方程，而是线性方程。

In IGCSE questions, you may see quadratics written in different orders or with brackets that need expanding first. Always rearrange the equation into the standard form before choosing a solution method.

在 IGCSE 题目中，二次方程可能以不同顺序出现，或者带有需要先展开的括号。选择解法前，一定要先把方程整理成标准形式。

ax² + bx + c = 0 (a ≠ 0)

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2. Standard Form and Key Terms | 标准形式与关键术语

fficient of x², b is the coefficient of x, and c is the constant term. The condition a ≠ 0 ensures that the equation really is quadratic.

二次方程的标准形式为 ax² + bx + c = 0。其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。条件 a ≠ 0 确保方程确实是二次方程。

alues correctly is important because they are used directly in the quadratic formula and the discriminant.

例如，在 3x² − 7x + 2 = 0 中，a = 3，b = −7，c = 2。正确识别这些值非常重要，因为二次公式和判别式会直接使用它们。

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3. Solving by Factorisation | 因式分解法

− 2)(x − 3) = 0. If the product of two factors is zero, at least one factor must be zero.

当二次方程能够分解成两个简单括号时，因式分解法通常最快。例如 x² − 5x + 6 = 0 可分解为 (x − 2)(x − 3) = 0。如果两个因式的乘积为零，那么至少有一个因式为零。

or x = 3. Always check your answers by substituting them back into the original equation.

由此得到 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。一定要把答案代回原方程进行检验。

instance, 2x² + 7x + 3 = 0 can be written as (2x + 1)(x + 3) = 0, giving x = −½ or x = −3.

当 a ≠ 1 时，可能需要拆中间项或尝试配对。例如，2x² + 7x + 3 = 0 可以写成 (2x + 1)(x + 3) = 0，得到 x = −½ 或 x = −3。

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4. Solving by Completing the Square | 配方法

ewrite ax² + bx + c = 0 in the form (x + p)² = q. Then take the square root of both sides.

配方法是一种适用于所有二次方程的可靠方法。其目标是将 ax² + bx + c = 0 改写为 (x + p)² = q 的形式，然后对两边开平方。

+ 6x + 2 = 0 can be written as (x + 3)² − 9 + 2 = 0, so (x + 3)² = 7. Therefore x + 3 = ±√7, giving x = −3 ± √7.

例如，x² + 6x + 2 = 0 可以写成 (x + 3)² − 9 + 2 = 0，所以 (x + 3)² = 7。因此 x + 3 = ±√7，得到 x = −3 ± √7。

If the coefficient a is not 1, divide the whole equation by a first. Then complete the square on the x² and x terms.

如果系数 a 不等于 1，先对方程整体除以 a，再对 x² 和 x 项进行配方。

x² + bx + c = 0 → (x + b/2)² − (b/2)² + c = 0

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5. Solving by the Quadratic Formula | 二次公式法

specially useful when the quadratic does not factorise easily. The formula is shown below.

二次公式可以直接给出 ax² + bx + c = 0 的解。当二次方程不容易因式分解时，它尤其有用。公式如下。

x = (−b ± √b² − 4ac) ÷ 2a

xample, solve 2x² − 3x − 5 = 0. Here a = 2, b = −3 and c = −5. Substituting gives x = (3 ± √9 + 40) ÷ 4 = (3 ± √49) ÷ 4 = (3 ± 7) ÷ 4, so x = 2.5 or x = −1.

例如，解 2x² − 3x − 5 = 0。这里 a = 2，b = −3，c = −5。代入公式得到 x = (3 ± √9 + 40) ÷ 4 = (3 ± √49) ÷ 4 = (3 ± 7) ÷ 4，所以 x = 2.5 或 x = −1。

Always write the formula with brackets around −b and the discriminant to avoid sign mistakes.

书写公式时，一定要在 −b 和判别式周围加上括号，避免符号错误。

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6. The Discriminant | 判别式

The discriminant is the expression b² − 4ac under the square root in the quadratic formula. It tells you how many real solutions a quadratic equation has before you solve it.

判别式是二次公式中平方根下的表达式 b² − 4ac。它可以在求解之前告诉你二次方程有多少个实数解。

• If b² − 4ac > 0, there are two distinct real solutions.

  如果 b² − 4ac > 0，方程有两个不同的实数解。

• If b² − 4ac = 0, there is exactly one repeated real solution.

  如果 b² − 4ac = 0，方程有一个重复的实数解。

• If b² − 4ac < 0, there are no real solutions, but two complex solutions at higher levels.

  如果 b² − 4ac < 0，方程没有实数解，但在更高阶段有两个复数解。

For example, x² + 4x + 5 = 0 has discriminant 16

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