Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. Solving quadratic equations is a core skill in IGCSE Mathematics, appearing in both Paper 2 and Paper 4. Mastery of this topic forms the foundation for graphs of functions, inequalities, and even introductory calculus in Additional Mathematics.

二次方程是最高次数为 2 的多项式方程，标准形式为 ax² + bx + c = 0，其中 a、b、c 是实数且 a ≠ 0。解二次方程是 IGCSE 数学的核心技能，在 Paper 2 和 Paper 4 中都会出现。掌握这一主题是学习函数图像、不等式乃至附加数学中微积分入门的基础。

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1. The Standard Form | 标准形式

Every quadratic equation can be rearranged into the standard form ax² + bx + c = 0. The coefficient a is the leading coefficient, b is the coefficient of x, and c is the constant term. If a = 0, the equation becomes linear, not quadratic, because the x² term disappears.

每个二次方程都可以整理成标准形式 ax² + bx + c = 0。其中 a 是二次项系数，b 是一次项系数，c 是常数项。如果 a = 0，由于 x² 项消失，方程就变成了线性方程，而不是二次方程。

For example, in 2x² − 3x + 1 = 0, we have a = 2, b = −3 and c = 1. The equation 5 − x² = 4x must be rearranged as x² + 4x − 5 = 0 before any solution method is applied. Remember: always bring all terms to one side and arrange them in descending powers of x.

例如，在 2x² − 3x + 1 = 0 中，a = 2，b = −3，c = 1。方程 5 − x² = 4x 必须整理成 x² + 4x − 5 = 0 后才能使用任何解法。记住：始终将所有项移到一侧，并按 x 的降幂排列。

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2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the quadratic has simple integer factors. The aim is to write ax² + bx + c as a product of two binomials, such as (px + q)(rx + s). Then use the zero product property: if the product of two factors is zero, at least one of the factors must be zero.

当二次式具有简单的整数因子时，因式分解是最快的方法。目标是将 ax² + bx + c 写成两个一次二项式的乘积，如 (px + q)(rx + s)。然后利用零积性质：如果两个因子的乘积为零，那么至少有一个因子为零。

Solve x² + 5x + 6 = 0. Find two numbers that multiply to c = 6 and add to b = 5, namely 2 and 3:

解 x² + 5x + 6 = 0。找出两个数相乘等于 c = 6 且相加等于 b = 5，即 2 和 3：

x² + 5x + 6 = (x + 2)(x + 3) = 0

x + 2 = 0 or x + 3 = 0

x = −2 or x = −3

Always expand your factorisation to check the answer: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. This verification takes only seconds and prevents careless errors.

务必展开因式以验证答案：(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6。这种验证只需几秒钟，可以避免粗心错误。

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3. Factorising Non-Monic Quadratics | 非首一二次式的因式分解

When a ≠ 1, factorisation requires more careful thinking. Consider 2x² + 7x + 3 = 0. Multiply a by c: 2 × 3 = 6. Now find two numbers that multiply to 6 and add to b = 7; these are 6 and 1. Rewrite the middle term and factor by grouping:

当 a ≠ 1 时，因式分解需要更细致的思考。考虑 2x² + 7x + 3 = 0。将 a 与 c 相乘：2 × 3 = 6。然后找出两个数相乘等于 6 且相加等于 b = 7，这两个数就是 6 和 1。重写中间项并分组因式分解：

2x² + 7x + 3 = 2x² + 6x + x + 3

= 2x(x + 3) + 1(x + 3)

= (2x + 1)(x + 3)

Hence (2x + 1)(x + 3) = 0, giving x = −1/2 or x = −3. This method, called the ‘ac method’, works for any factorisable quadratic. If the equation involves fractions, multiply through by the common denominator first.

因此 (2x + 1)(x + 3) = 0，得到 x = −1/2 或 x = −3。这种方法称为 ‘ac 法’，适用于任何可因式分解的二次式。如果方程中含有分数，先乘以公分母再求解。

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4. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q. This method always works and is essential for deriving the quadratic formula and for solving equations that do not factorise neatly.

配方法将 ax² + bx + c 改写为 a(x + p)² + q 的形式。这种方法始终有效，是推导求根公式和求解无法简洁因式分解的方程的关键。

Solve x² + 6x + 2 = 0. Take half of the coefficient of x, which is 3, and square it to obtain 9. Then add and subtract this value:

解 x² + 6x + 2 = 0。取 x 系数的一半，即 3，平方得到 9。然后加上再减去这个值：

x² + 6x + 9 − 9 + 2 = 0

(x + 3)² − 7 = 0

(x + 3)² = 7

x + 3 = ±√7

x = −3 ± √7

When the coefficient of x² is not 1, factor it out of the first two terms first. For 2x² − 4x − 6 = 0, write 2(x² − 2x) − 6 = 0, then complete the square inside the bracket: 2[(x − 1)² − 1] − 6 = 0.

当 x² 的系数不是 1 时，先从前两项中将其提出。对于 2x² − 4x − 6 = 0，写成 2(x² − 2x) − 6 = 0，然后在括号内配方：2[(x − 1)² − 1] − 6 = 0。

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5. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation and is derived by completing the square on the general equation ax² + bx + c = 0:

求根公式可以解任何二次方程，它通过对一般方程 ax² + bx + c = 0 配方推导得出：

x = (−b ± √(b² − 4ac)) / 2a

For the equation 2x² − 5x − 3 = 0, identify a = 2, b = −5, c = −3. Substitute these values carefully:

对于方程 2x² − 5x − 3 = 0，确定 a = 2，b = −5，c = −3。小心代入这些值：

x = (5 ± √((−5)² − 4 × 2 × (−3))) / (2 × 2)

x = (5 ± √(25 + 24)) / 4 = (5 ± √49) / 4 = (5 ± 7) / 4

x = 3 or x = −1/2

Note that (−5)² = 25, not −25. This is one of the most common sign errors in the exam. The formula gives exact answers when the square root is left in surd form; you may round to a required degree of accuracy if the question specifies it.

注意 (−5)² = 25，而不是 −25。这是考试中最常见的符号错误之一。当

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