Solving Quadratic Equations | 二次方程解法

📚 Solving Quadratic Equations | 二次方程解法

In the IGCSE Mathematics syllabus, quadratic equations are one of the most heavily tested topics in algebra. A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. This topic connects directly to factorisation, graph sketching, inequalities and problem solving, so mastering it will boost your marks across many exam questions.

在 IGCSE 数学考纲中，二次方程是代数部分考查频率最高的内容之一。任何可以写成 ax² + bx + c = 0 的方程都称为二次方程，其中 a、b、c 为常数，且 a ≠ 0。这一主题与因式分解、函数图像、不等式以及应用题紧密相连，因此掌握好二次方程能帮助你在多类考题中稳定拿分。

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1. The Standard Form of a Quadratic Equation | 二次方程的标准形式

Every quadratic equation can be rearranged into the standard form ax² + bx + c = 0. The coefficient a is called the leading coefficient, b is the linear coefficient, and c is the constant term. If a = 0, the equation becomes linear, which is why a must never be zero.

每个二次方程都可以整理成标准形式 ax² + bx + c = 0。系数 a 称为二次项系数，b 称为一次项系数，c 称为常数项。如果 a = 0，方程就退化为一次方程，因此 a 一定不能为零。

Before solving, always move every term to one side so that the right-hand side is zero. For example, 2x² = 5x − 3 must first be rewritten as 2x² − 5x + 3 = 0. Failing to rearrange the equation is one of the most common errors in the exam, so make this step a habit.

解方程之前，务必移项使所有项都在等号一边，另一边为零。例如，2x² = 5x − 3 必须先改写为 2x² − 5x + 3 = 0。忘记整理方程是考试中最常见的错误之一，请务必养成这个习惯。

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2. Solving by Factorisation | 因式分解法

Factorisation is the fastest method when the expression factorises cleanly. It relies on the zero-product property: if the product of two expressions is zero, then at least one of the expressions must be zero. In symbols, if AB = 0, then A = 0 or B = 0.

当表达式可以干净分解时，因式分解是最快的方法。它的依据是零积性质：如果两个表达式的乘积为零，那么至少其中一个表达式为零。用符号表示：若 AB = 0，则 A = 0 或 B = 0。

For the simple case x² + bx + c, look for two integers whose product is c and whose sum is b. For example, in x² − 5x + 6, we need two numbers that multiply to 6 and add to −5; these are −2 and −3.

对于简单的 x² + bx + c，我们需要找到两个整数，使它们的乘积等于 c，和等于 b。例如，在 x² − 5x + 6 中，我们需要两个相乘得 6、相加得 −5 的数，即 −2 和 −3。

Example: Solve x² − 5x + 6 = 0.

The factorisation is (x − 2)(x − 3) = 0. By the zero-product property, x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3. Always state both solutions clearly.

因式分解为 (x − 2)(x − 3) = 0。根据零积性质，x − 2 = 0 或 x − 3 = 0，因此 x = 2 或 x = 3。记得把两个解都写清楚。

When the leading coefficient a is not 1, factorise by grouping or by inspection. For example, 2x² + 7x + 3 = (2x + 1)(x + 3). Check by expanding: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. The check is quick and prevents careless mistakes.

当二次项系数 a ≠ 1 时，可以用分组分解法或直接观察法。例如，2x² + 7x + 3 = (2x + 1)(x + 3)。展开验证：(2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3。验证过程很快，能避免粗心错误。

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3. The Quadratic Formula | 求根公式

Not every quadratic expression factorises neatly, especially when the roots are irrational or involve fractions. In these cases, we use the quadratic formula, which works for every quadratic equation.

并非每个二次表达式都能干净地分解，尤其是当根为无理数或分数时。此时我们使用求根公式，它适用于所有二次方程。

For ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ⁄ 2a

Always write down the values of a, b and c before substituting. Then substitute them carefully into the formula and simplify. The ± symbol means you must calculate both the sum and the difference separately.

代入之前务必先写出 a、b、c 的值，然后小心代入公式并化简。± 符号表示你需要分别计算加法与减法两种情况。

Example: Solve 3x² + 5x − 2 = 0.

Here a = 3, b = 5, c = −2. Substituting into the formula gives x = (−5 ± √(5² − 4 × 3 × (−2))) ⁄ (2 × 3) = (−5 ± √(25 + 24)) ⁄ 6 = (−5 ± √49) ⁄ 6 = (−5 ± 7) ⁄ 6. Therefore x = 2 ⁄ 6 = 1 ⁄ 3, or x = −12 ⁄ 6 = −2.

这里 a = 3，b = 5，c = −2。代入公式得 x = (−5 ± √(5² − 4 × 3 × (−2))) ⁄ (2 × 3) = (−5 ± √(25 + 24)) ⁄ 6 = (−5 ± √49) ⁄ 6 = (−5 ± 7) ⁄ 6。因此 x = 2 ⁄ 6 = 1 ⁄ 3，或 x = −12 ⁄ 6 = −2。

Which method should you choose? Use this quick guide in the exam.

考试中应该选择哪种方法？请参考以下快速指南。

Method	Best used when
Factorisation	Roots are integers or simple fractions
Completing the square	You need the turning point or the equation is already in squared form
Quadratic formula	Factorisation is difficult or impossible

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4. Completing the Square | 配方法

Completing the square rewrites ax² + bx + c in the form p(x + q)² + r. This form is extremely useful because it directly gives the turning point of the parabola and can also be used to solve equations.

配方法将 ax² + bx + c 改写成 p(x + q)² + r 的形式。这种形式非常有用，因为它直接给出抛物线的顶点坐标，同时也可以用来解方程。

When a = 1, take half of the coefficient b, square it, and adjust the constant. For example, x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7. The turning point here is (−3, −7), and the line of symmetry is x = −3.

当 a = 1 时，取一次项系数 b 的一半并平方，再调整常数项。例如，x² + 6x + 2 = (x + 3)² − 9 + 2 = (x + 3)² − 7。此时顶点为 (−3, −7)，对称轴为 x = −3。

When a

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