Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. In the Edexcel IGCSE Mathematics course, quadratics appear in many different contexts: factorising, solving equations, sketching graphs, and solving real-life problems. Mastering this topic is essential for reaching the highest grades.

二次方程是指可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 为常数，且 a ≠ 0。在 Edexcel IGCSE 数学课程中，二次方程出现在许多不同的场景中：因式分解、解方程、绘制图像以及解决实际问题。掌握这一知识点是取得高分的关键。

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1. The Standard Form | 标准形式

Every quadratic equation can be rearranged into the form ax² + bx + c = 0. Here a is the coefficient of x², b is the coefficient of x, and c is the constant term. For example, 3x² − 5x + 2 = 0 has a = 3, b = −5 and c = 2.

每一个二次方程都可以整理为 ax² + bx + c = 0 的形式。其中 a 是 x² 的系数，b 是 x 的系数，c 是常数项。例如，3x² − 5x + 2 = 0 中 a = 3，b = −5，c = 2。

The values of x that satisfy the equation are called the roots or solutions of the equation. A quadratic equation can have two distinct real roots, one repeated root, or no real roots, depending on the value of the discriminant.

满足方程的 x 值称为方程的根或解。根据判别式的取值，二次方程可以有两个不同的实数根、一个重根，或者没有实数根。

Before solving, always check that the equation is in standard form and that a is not zero. If a = 0, the equation is linear, not quadratic.

在求解之前，务必检查方程是否已化为标准形式，并确认 a 不为零。若 a = 0，则方程为一次方程而非二次方程。

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2. Solving by Factorisation | 因式分解法

Factorisation is the quickest method when the quadratic has simple integer roots. The zero product property states that if the product of two expressions is zero, then at least one of the expressions must be zero.

当二次方程具有简单的整数根时，因式分解是最快捷的方法。零积性质指出：若两个表达式的乘积为零，则其中至少有一个表达式为零。

Example: solve x² − 5x + 6 = 0. We look for two numbers that multiply to give 6 and add to give −5. These numbers are −2 and −3, so we write (x − 2)(x − 3) = 0. Therefore x = 2 or x = 3.

例：解 x² − 5x + 6 = 0。我们需要找到两个数，它们相乘为 6，相加为 −5。这两个数是 −2 和 −3，因此可以写成 (x − 2)(x − 3) = 0。所以 x = 2 或 x = 3。

For quadratics where the coefficient of x² is not 1, factorisation requires more care. For example, 2x² + 5x + 3 = 0 can be written as (2x + 3)(x + 1) = 0, giving x = −3/2 or x = −1.

当 x² 的系数不为 1 时，因式分解需要更加细心。例如，2x² + 5x + 3 = 0 可以写成 (2x + 3)(x + 1) = 0，从而得到 x = −3/2 或 x = −1。

Always expand your brackets to check your factorisation before solving. This simple step prevents many careless errors in the examination.

在求解之前，一定要展开括号以检查因式分解是否正确。这一简单步骤可以避免考试中的许多粗心错误。

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3. Solving by Completing the Square | 配方法

Completing the square rewrites a quadratic in the form (x + p)² − q. This is especially useful when the equation cannot be factorised easily, and it is also the key to finding turning points of quadratic graphs.

配方法将二次式改写为 (x + p)² − q 的形式。当方程不易因式分解时，这种方法特别有用，同时也是求二次函数图像顶点坐标的关键。

Example: solve x² + 6x − 7 = 0. First halve the coefficient of x, which is 6, to get 3. Write (x + 3)² − 9 − 7 = 0, so (x + 3)² − 16 = 0. Then (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = −7.

例：解 x² + 6x − 7 = 0。首先将 x 的系数 6 除以 2，得到 3。写出 (x + 3)² − 9 − 7 = 0，即 (x + 3)² − 16 = 0。于是 (x + 3)² = 16，所以 x + 3 = ±4，得到 x = 1 或 x = −7。

When the coefficient of x² is not 1, factor that coefficient out first. For 2x² + 8x + 3 = 0, write 2[x² + 4x] + 3 = 0 and then complete the square inside the bracket: 2[(x + 2)² − 4] + 3 = 0, giving 2(x + 2)² − 5 = 0.

当 x² 的系数不为 1 时，先将该系数提取出来。对于 2x² + 8x + 3 = 0，先写成 2[x² + 4x] + 3 = 0，再在括号内配方：2[(x + 2)² − 4] + 3 = 0，即 2(x + 2)² − 5 = 0。

Remember that the final answer from completing the square often contains surds. In Edexcel IGCSE examinations, leaving your answer in surd form is usually acceptable and sometimes required.

请记住，配方法得出的答案常常含有根式。在 Edexcel IGCSE 考试中，答案保留根式形式通常是可以接受的，有时甚至是题目要求的。

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4. The Quadratic Formula | 二次公式

For any quadratic equation ax² + bx + c = 0, the solutions are given by the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，其解由二次公式给出：

x = (−b ± √(b² − 4ac)) / 2a

This formula works for every quadratic, including those that cannot be factorised. You do not need to memorise the derivation, but you must be able to substitute values correctly and simplify carefully.

该公式适用于所有二次方程，包括无法因式分解的方程。你不需要记住推导过程，但必须能够正确代入数值并仔细化简。

Example: solve 2x² + 3x − 5 = 0. Substituting a = 2, b = 3 and c = −5 gives:

例：解 2x² + 3x − 5 = 0。将 a = 2，b = 3，c = −5 代入得：

x = (−3 ± √(3² − 4 × 2 × (−5))) / (2 × 2) = (−3 ± √49) / 4

Therefore x = (−3 + 7) / 4 = 1, or x = (−3 − 7) / 4 = −5/2.

因此 x = (−3 + 7) / 4 = 1，或 x = (−3 − 7) / 4 = −5/2。

Take care with negative signs inside the square root. It is a common mistake to write b² − 4ac incorrectly when c is negative,

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