Quadratic Equations: Factorising, Completing the Square, and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

📚 Quadratic Equations: Factorising, Completing the Square, and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式

Quadratic equations are a central topic in IGCSE Mathematics. They appear in algebra, graphs, and real-life problems. A quadratic equation can have two solutions, one solution, or no real solutions. This article revises the main methods: factorising, completing the square, and using the quadratic formula.

二次方程是 IGCSE 数学的核心主题之一。它们出现在代数、函数图像和实际应用问题中。一个二次方程可能有两个解、一个解或没有实数解。本文复习三种主要解法：因式分解法、配方法和求根公式法。

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

usually be written in the form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.

二次方程是指未知数的最高次数为 2 的方程。它通常可以写成 ax² + bx + c = 0 的形式，其中 a、b、c 是常数，且 a ≠ 0。

0, the equation becomes linear, not quadratic. The condition a ≠ 0 is therefore essential.

如果 a = 0，方程就变成了一次方程，而不是二次方程。因此条件 a ≠ 0 非常重要。

Examples include x² - 5x + 6 = 0, 2x² + 3x - 2 = 0, and x² - 9 = 0.

例如 x² - 5x + 6 = 0、2x² + 3x - 2 = 0 和 x² - 9 = 0 都是二次方程。

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2. Standard Form ax² + bx + c = 0 | 标准形式 ax² + bx + c = 0

ndard form: ax² + bx + c = 0. All terms must be on one side, and the other side must be 0.

在求解之前，必须先把方程整理成标准形式：ax² + bx + c = 0。所有项都要移到一边，另一边必须为 0。

example, x² + 4 = -4x should be rearranged to x² + 4x + 4 = 0 by adding 4x to both sides.

例如，x² + 4 = -4x 应在两边加上 4x，整理为 x² + 4x + 4 = 0。

form allows you to identify a, b and c correctly. In x² + 4x + 4 = 0, a = 1, b = 4, c = 4.

标准形式能帮助你正确识别 a、b、c。在 x² + 4x + 4 = 0 中，a = 1，b = 4，c = 4。

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3. Solving by Factorising | 因式分解法求解

r factors. You look for two numbers that multiply to give ac and add to give b when a = 1.

当二次式能写成两个一次因式的乘积时，因式分解法是最快的方法。当 a = 1 时，需要找到两个数，使它们的乘积等于 ac，和等于 b。

Example: Solve x² - 7x + 12 = 0.

例题：解方程 x² - 7x + 12 = 0。

s whose product is 12 and sum is -7. They are -3 and -4, so x² - 7x + 12 = (x - 3)(x - 4).

需要两个数，乘积为 12，和为 -7。这两个数是 -3 和 -4，因此 x² - 7x + 12 = (x - 3)(x - 4)。

The equation becomes (x - 3)(x - 4) = 0.

方程变为 (x - 3)(x - 4) = 0。

Using the zero product property, x - 3 = 0 or x - 4 = 0, so x = 3 or x = 4.

利用零积性质，x - 3 = 0 或 x - 4 = 0，所以 x = 3 或 x = 4。

When a ≠ 1, use the ac method or trial and error to split the middle term.

当 a ≠ 1 时，可用 ac 法或试错法拆分中间项。

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4. The Zero Product Property | 零积性质

The zero product property states that if the product of two expressions is zero, then at least one of the expressions must be zero. This is why we set each factor equal to zero.

零积性质指出，如果两个式子的乘积为零，那么其中至少有一个式子必须为零。这就是我们把每个因式分别设为零的原因。

For (x + 2)(x - 5) = 0, we write x + 2 = 0 or x - 5 = 0, giving x = -2 or x = 5.

对于 (x + 2)(x - 5) = 0，我们写 x + 2 = 0 或 x - 5 = 0，得到 x = -2 或 x = 5。

is 0. If the equation is (x + 2)(x - 5) = 6, you cannot simply set the factors equal to 6.

这个性质只有在等式一边为 0 时才成立。如果方程是 (x + 2)(x - 5) = 6，不能简单地把因式分别设为 6。

In that case, expand, rearrange to standard form, and then factorise.

此时应展开、整理成标准形式，然后再因式分解。

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5. Completing the Square | 配方法

Completing the square rewrites a quadratic expression in the form (x + p)² + q. This is useful when factorising is difficult or when finding the vertex of a graph.

配方法把二次式写成 (x + p)² + q 的形式。当因式分解困难或需要求图像顶点时，这种方法很有用。

take half of b, square it, and adjust the constant: x² + bx + c = (x + b/2)² - (b/2)² + c.

对于 x² + bx + c，取 b 的一半并平方，再调整常数：x² + bx + c = (x + b/2)² - (b/2)² + c。

Example: Solve x² + 6x + 5 = 0 by completing the square.

例题：用配方法解方程 x² + 6x + 5 = 0。

Half of 6 is 3, and 3² = 9, so x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4.

6 的一半是 3，3² = 9，所以 x² + 6x + 5 = (x + 3)² - 9 + 5 = (x + 3)² - 4。

Set (x + 3)² - 4 = 0, so (x + 3)² = 4.

令 (x + 3)² - 4 = 0，则 (x + 3)² = 4。

Taking square roots gives x + 3 = ±2, so x = -3 ± 2, giving x = -1 or x = -5.

两边开平方得 x + 3 = ±2，所以 x = -3 ± 2，解得 x = -1 或 x = -5。

When the coefficient of x² is not 1, factor it out first before completing the square.

当 x² 的系数不是 1 时，先提出该系数，再进行配平方。

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6. The Quadratic Formula | 求根公式

The quadratic formula solves any quadratic equation ax² + bx + c = 0. It is:

求根公式可以解任意二次方程 ax² + bx + c = 0。公式如下：

x = (-b ± √b² - 4ac) / 2a

This formula comes from completing the square on the general quadratic equation.

这个公式是由对一般二次方程进行配平方推导而来。

Example: Solve 2x² - 4x - 6 = 0 using the quadratic formula.

例题：用求根公式解方程 2x² - 4x - 6 = 0。

Here a = 2, b = -4, c = -6. Substitute: x = (-(-4) ± √((-4)² - 4 × 2 × (-6))) / (2 × 2).

这里 a = 2，b = -4，c = -6。代入公式：x = (4 ± √16 + 48) / 4 = (4 ± √64) / 4。

So x = (4 ± 8) / 4, giving x = 12 / 4 = 3 or x = -4 / 4 = -1.

因此 x = (4 ± 8) / 4，解得 x = 12 / 4 = 3 或 x = -4 / 4 = -1。

The quadratic formula is especially useful when the quadratic does not factorise easily.

当二次式不容易因式分解时，求根公式尤其有用。

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7. The Discriminant | 判别式

The discriminant is the expression under the square root in the quadratic formula:

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