📚 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 appear throughout the IGCSE Mathematics syllabus, from algebraic manipulation to graphs, problem solving, and even geometry. This revision guide explains the three main algebraic methods for solving quadratic equations: factorising, completing the square, and using the quadratic formula. It also covers the discriminant, common exam mistakes, and practice questions to strengthen your understanding.
二次方程贯穿 IGCSE 数学大纲，从代数运算到函数图像、应用题甚至几何问题都有涉及。本篇复习指南讲解求解二次方程的三种主要代数方法：因式分解法、配方法和求根公式法，同时涵盖判别式、常见考试错误和练习题，帮助你巩固理解。
1. What is a Quadratic Equation? | 什么是二次方程？
a, b and c are constants and a ≠ 0. If a = 0, the equation becomes linear, not quadratic.
二次方程是指未知数的最高次数为 2 的方程。其标准形式为 ax² + bx + c = 0，其中 a、b、c 为常数且 a ≠ 0。如果 a = 0，方程就变成了一次方程，而不是二次方程。
arabola, and the solutions correspond to the points where the parabola crosses the x-axis.
二次方程可能有两个实数解、一个重根实数解，或没有实数解。函数 y = ax² + bx + c 的图像是一条抛物线，方程的解对应抛物线与 x 轴交点的横坐标。
two solutions, x = 2 and x = 3, because both values make the left-hand side equal to zero.
例如，x² − 5x + 6 = 0 是一个二次方程，因为 x 的最高次数是 2。它有两个解，x = 2 和 x = 3，因为这两个值都能使等号左边等于 0。
2. Standard Form and Key Terms | 标准形式与关键术语
efficient a is the number before x², b is the number before x, and c is the constant term.
在求解二次方程之前，一定要先把它整理成 ax² + bx + c = 0 的形式。例如，3x² = 5x − 2 应写成 3x² − 5x + 2 = 0。系数 a 是 x² 前的数，b 是 x 前的数，c 是常数项。
ation on both sides. In 2x² + x = 6, subtract 6 from both sides to obtain 2x² + x − 6 = 0.
整理时，要把所有项移到方程的一边并合并同类项。方程两边要同时进行相同的运算以保持平衡。例如 2x² + x = 6，两边减去 6，得到 2x² + x − 6 = 0。
Key terms include root, solution, zero, and x-intercept. These all refer to a value of x that makes the equation true. In the quadratic formula, the expression b² − 4ac is called the discriminant and is written as Δ.
关键术语包括根、解、零点和 x 轴截距，它们都指使方程成立的 x 值。在求根公式中，表达式 b² − 4ac 称为判别式，记作 Δ。
3. Solving by Factorising | 因式分解法
. This is called the zero product property, and it gives two simple linear equations to solve.
当二次式可以写成两个一次因式的乘积时，可以使用因式分解法。如果 (px + q)(rx + s) = 0，则 px + q = 0 或 rx + s = 0。这称为零乘积性质，由此得到两个简单的一次方程来求解。
7x + 10 = 0 factors as (x − 2)(x − 5) = 0 because −2 and −5 multiply to 10 and add to −7.
对于 x² + bx + c 形式的二次式，找出两个数，使它们的乘积等于 c，和等于 b。例如，x² − 7x + 10 = 0 可以分解为 (x − 2)(x − 5) = 0，因为 −2 和 −5 的乘积是 10，和是 −7。
For ax² + bx + c where a ≠ 1, either use trial and error or split the middle term. The splitting method rewrites bx as the sum of two terms whose coefficients multiply to ac and add to b.
对于 a ≠ 1 的 ax² + bx + c 形式，可使用试错法或拆中项法。拆中项法把 bx 拆成两项之和，使这两项的系数乘积等于 ac，和等于 b。
4. Worked Example: Factorising | 因式分解法例题
5x + 6 = (x + 2)(x + 3) = 0. Therefore x + 2 = 0 gives x = −2, and x + 3 = 0 gives x = −3.
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图