Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与二次公式

📚 Quadratic Equations: Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与二次公式

In IGCSE Mathematics, quadratic equations appear in algebra, graphs, geometry and even word problems. 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 revision guide explains three core methods: factorising, completing the square and using the quadratic formula. You will also learn how the discriminant helps predict the number of roots.

在 IGCSE 数学中，二次方程广泛出现在代数、图像、几何甚至应用题中。二次方程是指任何可以写成 ax² + bx + c = 0 形式的方程，其中 a、b、c 为常数且 a ≠ 0。本复习指南将讲解三种核心方法：因式分解法、配方法和二次公式法。你还将学习判别式如何帮助判断根的个数。

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

A quadratic equation is a polynomial equation of degree 2. The highest power of the unknown x is 2. Its standard form is ax² + bx + c = 0, where a, b and c are real numbers and a cannot be zero. If a = 0, the equation becomes linear.

二次方程是一个次数为 2 的多项式方程，未知数 x 的最高次数是 2。它的标准形式为 ax² + bx + c = 0，其中 a、b、c 为实数且 a 不能为 0。如果 a = 0，方程就变成了一次方程。

Examples include x² − 5x + 6 = 0, 2x² + 3x − 2 = 0 and x² − 4 = 0. Some equations may need rearranging before they match the standard form.

例如 x² − 5x + 6 = 0、2x² + 3x − 2 = 0 和 x² − 4 = 0。有些方程需要先整理才能变成标准形式。

ax² + bx + c = 0, a ≠ 0

A quadratic equation can have two real roots, one repeated real root or no real roots. The graph of y = ax² + bx + c is a parabola.

二次方程可以有两个实数根、一个重根实数根或没有实数根。y = ax² + bx + c 的图像是一条抛物线。

2. Standard Form and Coefficients | 标准形式与系数

To use any method, first write the equation in standard form ax² + bx + c = 0. Identify the coefficients a, b and c carefully, including their signs. For example, in 3x² − 7x + 2 = 0, a = 3, b = −7 and c = 2.

使用任何方法之前，先把方程写成标准形式 ax² + bx + c = 0。仔细确定系数 a、b 和 c，包括它们的符号。例如，在 3x² − 7x + 2 = 0 中，a = 3，b = −7，c = 2。

If the equation is given as 5x² = 2x + 1, rearrange it to 5x² − 2x − 1 = 0. Then a = 5, b = −2 and c = −1. A common mistake is to take b as +2 or c as +1.

如果题目给出 5x² = 2x + 1，应移项得 5x² −

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