📚 Quadratic Equations: From Factorising to the Quadratic Formula | 二次方程：从因式分解到求根公式
In IGCSE Mathematics, quadratic equations appear in algebra, graphs and problem solving. A solid understanding of factorising, completing the square and the quadratic formula helps you choose the most efficient method under exam conditions. This revision guide explains each technique step by step and highlights common errors to avoid.
在 IGCSE 数学中，二次方程广泛出现在代数、图像和实际问题中。扎实掌握因式分解、配方法和二次公式，能够帮助你在考试中选择最高效的解法。本复习指南将逐步讲解每种方法，并强调需要避免的常见错误。
1. What Is a Quadratic Equation? | 什么是二次方程？
ants and a ≠ 0. The highest power of the unknown x is 2, which is why it is called quadratic.
二次方程是可以写成一般形式 ax² + bx + c = 0 的方程，其中 a、b 和 c 是常数，且 a ≠ 0。未知数 x 的最高次数是 2，因此称为二次方程。
of x is 2. The equation x³ + 2x² − 1 = 0 is not quadratic because it contains an x³ term.
例如，2x² − 5x + 3 = 0 是二次方程，因为 x 的最高次数是 2。方程 x³ + 2x² − 1 = 0 不是二次方程，因为它含有 x³ 项。
In IGCSE exams, questions may give equations in different forms, so you often need to rearrange terms before solving. Some word problems also lead to quadratic equations after forming an algebraic model.
在 IGCSE 考试中，题目可能以不同形式给出方程，因此解题前通常需要先整理各项。一些应用题在建立代数模型后也会得到二次方程。
2. Standard Form and Key Vocabulary | 标准形式与关键术语
terms to one side and set the other side equal to zero. This makes factorising and formula methods easier to apply correctly.
解题前应先把方程写成标准形式 ax² + bx + c = 0。将所有项移到一边，令另一边等于 0。这样更容易正确应用因式分解法和公式法。
Key terms include the leading coefficient a, the coefficient b of x, the constant term c, and the roots or solutions. A root is any value of x that makes ax² + bx + c equal zero.
关键术语包括 x² 的系数 a、x 的系数 b、常数项 c，以及根或解。根是使 ax² + bx + c 等于零的任何 x 值。
The discriminant is b² − 4ac. It tells you the nature of the roots without solving the equation fully. You will use this concept often in IGCSE questions about equal roots or no real roots.
判别式是 b² − 4ac。它可以在不完整求解方程的情况下判断根的性质。在处理 IGCSE 中关于相等实根或无实根的问题时，你会经常用到这个概念。
3. Solving by Factorising | 因式分解法
he goal is to write ax² + bx + c as a product of two linear factors: (px + q)(rx + s) = 0.
当二次方程有有理根且系数 a 为 1 或较小整数时，因式分解通常是最快的方法。目标是将 ax² + bx + c 写成两个一次因式的乘积：(px + q)(rx + s) = 0。
12 and add to −7. The numbers are −3 and −4, so the factorised form is (x − 3)(x − 4) = 0.
例如，解方程 x² − 7x + 12 = 0。我们需要两个数，它们相乘得 12，相加得 −7。这两个数是 −3 和 −4，因此因式分解形式为 (x − 3)(x − 4) = 0。
+ 12 = x² − 7x + 12, which is correct. Always expand mentally or on paper if time allows.
你可以通过展开括号来检验：x² − 4x − 3x + 12 = x² − 7x + 12，这是正确的。如果时间允许，尽量在心里或在纸上展开检验。
0 because 2x × x = 2x² and 1 × 3 = 3 while the outer and inner products give 6x + x = 7x.
当 a ≠ 1 时，可使用拆中间项或试错法。例如，2x² + 7x + 3 = 0 可因式分解为 (2x + 1)(x + 3) = 0，因为 2x × x = 2x²，1 × 3 = 3，而外项和内项乘积之和为 6x + x = 7x。
4. The Zero Product Property | 零乘积性质
After factorising a quadratic into two factors, apply the zero product property: if a product equals zero, then at least one factor must equal zero. This allows you to split the equation into two simple linear equations.
将二次方程因式分解后，应用零乘积性质：如果乘积为零，那么至少有一个因式为零。这使你可以将方程拆分成两个简单的一次方程。
either x − 3 = 0 or x − 4 = 0, giving x = 3 or x = 4. Always state both solutions clearly.
由 (x − 3)(x − 4) = 0，可得 x − 3 = 0 或 x − 4 = 0，因此 x = 3 或 x = 4。始终要清晰地写出两个解。
range to x(x − 2) − x = 0, then factorise as x(x − 3) = 0 and solve to get x = 0 or x = 3.
一个常见错误是两边同时除以含 x 的因式，这可能会丢失解。例如，从 x(x − 2) = x 出发，不要除以 x。而应整理为 x(x − 2) − x = 0，再因式分解为 x(x − 3) = 0，解得 x = 0 或 x = 3。
Using the zero product property is powerful because it turns a quadratic problem into two linear ones. It only works when the product equals zero, so ensure the right-hand side is zero first.
零乘积性质非常有效，因为它把二次问题转化为两个一次问题。它只在乘积等于零
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