Solving Quadratic Equations: Factorisation, Formula & Completing the Square | 解二次方程：因式分解、公式法与配方法

📚 Solving Quadratic Equations: Factorisation, Formula & Completing the Square | 解二次方程：因式分解、公式法与配方法

A quadratic equation is a second-degree polynomial equation in one variable. It can always be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. A value of x that makes the equation true is called a root or a solution.

二次方程是只含一个变量、最高次数为2的多项式方程。它总可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 为常数，且 a ≠ 0。能使方程成立的 x 值称为方程的根或解。

ax² + bx + c = 0, a ≠ 0

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

Before solving any quadratic equation, it is essential to rearrange it into the standard form ax² + bx + c = 0. This means all terms must be on one side of the equal sign, with zero on the other side. The letters a, b and c are constants, and a cannot be zero; if a = 0, the equation becomes linear, not quadratic.

在解任何二次方程之前，必须先将其整理成标准形式 ax² + bx + c = 0。也就是说，把所有项移到等号的一边，使另一边为0。字母 a、b、c 是常数，且 a 不能为0；如果 a = 0，方程就变成了一次方程，而不是二次方程。

For example, the equation 2x² + 4x = 8 can be rearranged as 2x² + 4x − 8 = 0. In this case a = 2, b = 4 and c = −8.

例如，方程 2x² + 4x = 8 可以整理成 2x² + 4x − 8 = 0。此时 a = 2，b = 4，c = −8。

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2. Factorising Monic Quadratics (a = 1) | 分解首项系数为1的二次式

A monic quadratic is one where a = 1, so it has the form x² + bx + c. To factorise it, find two integers p and q such that p + q = b and p × q = c. Then the quadratic can be written as (x + p)(x + q).

首项系数为1的二次式形如 x² + bx + c。要分解它，需要找到两个整数 p 和 q，使 p + q = b，且 p × q = c。于是该二次式可以写成 (x + p)(x + q)。

Example 1: Factorise x² + 5x + 6. We need p + q = 5 and p × q = 6. The pair 2 and 3 works, so x² + 5x + 6 = (x + 2)(x + 3).

例1：分解 x² + 5x + 6。需要 p + q = 5 且 p × q = 6。2 和 3 满足条件，所以 x² + 5x + 6 = (x + 2)(x + 3)。

Example 2: Factorise x² − 2x − 8. Here p + q = −2 and p × q = −8. The pair 2 and −4 works, so x² − 2x − 8 = (x + 2)(x − 4).

例2：分解 x² − 2x − 8。这里 p + q = −2 且 p × q = −8。2 和 −4 满足条件，所以 x² − 2x − 8 = (x + 2)(x − 4)。

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3. Factorising Non-Monic Quadratics (a ≠ 1) | 分解首项系数不为1的二次式

When a is not 1, use the product-sum method. First multiply a and c to get ac. Then find two numbers m and n such that m + n = b and m × n = ac. Split the middle term bx into mx + nx, and finish by factoring by grouping.

当 a 不等于1时，可以使用“乘积-和”法。先把 a 和 c 相乘得到 ac。然后找两个数 m 和 n，使 m + n = b，且 m × n = ac。把中间项 bx 拆成 mx + nx，再用分组分解法完成。

Example: Factorise 2x² + 7x + 3. Here a × c = 6 and b = 7, so m = 6 and n = 1. Rewrite the expression as 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).

例：分解 2x² + 7x + 3。这里 a × c = 6，b = 7，所以 m = 6，n = 1。原式改写为 2x² + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)。

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4. Solving by Factorisation | 用因式分解法求解

To solve a quadratic equation by factorisation, first rewrite the equation in the form ax² + bx + c = 0. Then factorise the left side completely and use the zero-product property: if A × B = 0, then A = 0 or B = 0.

用因式分解法解二次方程时，先把方程化为 ax² + bx + c = 0 的形式，然后完全分解左边，再利用零积性质：若 A × B = 0，则 A = 0 或 B = 0。

Example: Solve x² − 5x + 6 = 0. Factorise: (x − 2)(x − 3) = 0. The zero-product property gives x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.

例：解方程 x² − 5x + 6 = 0。分解得 (x − 2)(x − 3) = 0。由零积性质得 x − 2 = 0 或 x − 3 = 0，所以 x = 2 或 x = 3。

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5. The Quadratic Formula | 二次求根公式

For any quadratic equation ax² + bx + c = 0, the roots can be found directly using the quadratic formula:

对于任意二次方程 ax² + bx + c = 0，都可以直接用求根公式求出根：

x = (−b ± √(b² − 4ac)) / (2a)

Example: Solve 2x² + 3x − 2 = 0. Here a = 2, b = 3 and c = −2. Substitute into the formula:

例：解方程 2x² + 3x − 2 = 0。这里 a = 2，b = 3，c = −2。代入公式：

x = (−3 ± √(9 − 4 × 2 × (−2))) / (2 × 2) = (−3 ± √25) / 4 = (−3 ± 5) / 4

So x = (−3 + 5) / 4 = 1/2 or x = (−3 − 5) / 4 = −2.

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

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6. Deriving the Formula by Completing the Square | 用配方法推导公式

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