Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | IGCSE数学：二次方程求解（因式分解法、配方法与二次公式）

📚 Solving Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | IGCSE数学：二次方程求解（因式分解法、配方法与二次公式）

Quadratic equations appear throughout IGCSE algebra, from factorising simple trinomials to solving problems involving areas, projectiles and financial models. This revision guide explains three core methods: factorising, completing the square and using the quadratic formula. It also covers the discriminant, graphical interpretation, word problems, common mistakes and exam techniques.

二次方程贯穿 IGCSE 代数内容，从简单的三项式因式分解到涉及面积、抛体运动和金融模型的实际问题。本复习指南讲解三种核心方法：因式分解法、配方法和二次公式法。同时涵盖判别式、图像解释、应用题、常见错误和考试技巧。

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

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

二次方程是任何可以写成标准形式 ax² + bx + c = 0 的方程，其中 a、b、c 是常数且 a ≠ 0。

on a ≠ 0 is essential because if a = 0, the equation becomes linear rather than quadratic.

条件 a ≠ 0 非常重要，因为如果 a = 0，方程就变成一次方程而不是二次方程。

For example, 2x² + 3x - 5 = 0, x² - 9 = 0 and -x² + 4x = 0 are all quadratic equations.

例如，2x² + 3x - 5 = 0、x² - 9 = 0 和 -x² + 4x = 0 都是二次方程。

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2. Solving by Factorising | 因式分解法

Factorising is often the fastest method when a quadratic expression can be written as a product of two linear factors.

当二次式可以写成两个一次因式的乘积时，因式分解法通常是最快的方法。

The standard steps are: first set the equation equal to zero, then factorise the left-hand side, set each factor equal to zero, and finally solve the resulting linear equations.

标准步骤是：先将方程设为零，然后对左侧进行因式分解，令每个因式为零，最后解出相应的一次方程。

Example: solve x² - 7x + 12 = 0.

例如：解 x² - 7x + 12 = 0。

x² - 7x + 12 = (x - 3)(x - 4) = 0, so x = 3 or x = 4.

x² - 7x + 12 = (x - 3)(x - 4) = 0，因此 x = 3 或 x = 4。

mmon factor first. For example, 2x² - 8x = 0 becomes 2x(x - 4) = 0, giving x = 0 or x = 4.

始终先寻找公因式。例如，2x² - 8x = 0 变为 2x(x - 4) = 0，得到 x = 0 或 x = 4。

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3. Solving by Completing the Square | 配方法

Completing the square is useful when a quadratic expression does not factorise easily, and it also helps to find the vertex of a parabola.

当二次式不容易因式分解时，配方法非常有用，它也有助于求出抛物线的顶点。

equation of the form x² + bx + c = 0, first move the constant term to the right-hand side.

对于形如 x² + bx + c = 0 的方程，首先将常数项移到等式右边。

Then add (b ÷ 2)² to both sides, which creates a perfect square on the left.

然后在两边同时加上 (b ÷ 2)²，从而在左边形成一个完全平方。

Example: solve x² + 6x - 7 = 0.

例如：解 x² + 6x - 7 = 0。

x² + 6x = 7, then x² + 6x + 9 = 7 + 9, so (x + 3)² = 16.

x² + 6x = 7，然后 x² + 6x + 9 = 7 + 9，所以 (x + 3)² = 16。

Taking square roots gives x + 3 = ±4, so x = 1 or x = -7.

开方后得到 x + 3 = ±4，因此 x = 1 或 x = -7。

If the coefficient a is not 1, divide every term by a before completing the square.

如果系数 a 不等于 1，在配方之前先将每一项都除以 a。

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4. The Quadratic Formula | 二次公式

The quadratic formula can solve any quadratic equation in standard form, even when factorising is not possible.

二次公式可以解标准形式的任何二次方程，即使无法因式分解也能使用。

x = (-b ± √b² - 4ac) / 2a

his formula is obtained by completing the square on the general equation ax² + bx + c = 0.

该公式是通过对一般方程 ax² + bx + c = 0 进行配方得到的。

Example: solve 2x² - 4x - 6 = 0. Here a = 2, b = -4 and c = -6, so the formula gives:

例如：解 2x² - 4x - 6 = 0。这里 a = 2、b = -4、c = -6，代入公式得到：

x = (4 ± √16 + 48) / 4 = (4 ± 8) / 4, so x = 3 or x = -1.

x = (4 ± √16 + 48) / 4 = (4 ± 8) / 4，因此 x = 3 或 x = -1。

When using the formula, be careful with signs, especially when b is negative or when substituting c correctly.

使用公式时要特别注意符号，尤其是当 b 为负数或代入 c 时容易出错。

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5. The Discriminant | 判别式

The discriminant of a quadratic equation ax² + bx + c = 0 is given by Δ = b² - 4ac.

二次方程 ax² + bx + c = 0 的判别式由 Δ = b² - 4ac 给出。

It tells you the nature of the roots without actually solving the equation.

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