Solving Quadratic Equations | 解二次方程

📚 Solving Quadratic Equations | 解二次方程

A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. In the Edexcel IGCSE Mathematics specification, quadratic equations are examined in both Foundation and Higher tier papers, and they appear in almost every session. You will be expected to solve them confidently and efficiently, whether the question is purely algebraic or embedded in a word problem.

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 为常数且 a ≠ 0。在 Edexcel IGCSE 数学考纲中，二次方程在基础卷和提高卷中都会考查，几乎每次考试都会出现。无论是纯代数题目还是来自实际应用题的设问，你都需要自信而高效地求解。

1. The Standard Form | 标准形式

Before attempting any solution method, the equation must first be rearranged into the standard form ax² + bx + c = 0. This means moving all terms to one side, so the other side is zero. The coefficient a must never be zero, because if a = 0 then the x² term disappears and the equation becomes linear. For example, 3x² − 5x + 2 = 0 is a quadratic, while 2x − 7 = 0 is a linear equation.

在尝试任何解法之前，必须先将方程整理成标准形式 ax² + bx + c = 0。这意味着把所有项移到一边，使另一边为零。系数 a 绝不能为零，因为若 a = 0，x² 项就消失了，方程就变成一次方程。例如，3x² − 5x + 2 = 0 是二次方程，而 2x − 7 = 0 是一次方程。

Two special cases deserve attention. If c = 0, the equation has the form ax² + bx = 0, and the common factor x can be extracted immediately. If b = 0, the equation has the form ax² + c = 0, and it can be solved by isolating x² and taking square roots.

有两种特殊情况值得注意。若 c = 0，方程形如 ax² + bx = 0，可以直接提取公因式 x。若 b = 0，方程形如 ax² + c = 0，可以通过单独留下 x² 再取平方根来求解。

• For ax² + bx = 0: factorise as x(ax + b) = 0.

  对于 ax² + bx = 0：分解为 x(ax + b) = 0。

• For ax² + c = 0: x² = −c/a, then x = ±√(−c/a).

  对于 ax² + c = 0：x² = −c/a，然后 x = ±√(−c/a)。

2. Expanding Double Brackets | 展开双括号

To solve a quadratic by factorisation, you must be completely comfortable with expanding and factorising double brackets. Expanding removes the brackets: (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10. Use the FOIL order: First, Outer, Inner, Last, and then combine the like terms.

要用因式分解法解二次方程，你必须完全熟练展开和分解双括号。展开就是去括号：(x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10。按照 FOIL 顺序：首项、外项、内项、末项，然后合并同类项。

Factorising is the reverse process. To factorise x² + 7x + 10, find two numbers that multiply to give the constant term 10 and add to give the coefficient of x, which is 7. The numbers 2 and 5 satisfy both conditions, so x² + 7x + 10 = (x + 2)(x + 5). Always check your factorisation by expanding it back.

因式分解是展开的逆过程。要分解 x² + 7x + 10，找到两个数相乘等于常数项 10，相加等于 x 的系数 7。数字 2 和 5 同时满足两个条件，所以 x² + 7x + 10 = (x + 2)(x + 5)。务必通过重新展开来检查你的分解是否正确。

3. Difference of Two Squares and Perfect Squares | 平方差与完全平方

The difference of two squares is a special pattern that appears very frequently in IGCSE papers: a² − b² = (a + b)(a − b). For example, x² − 16 = (x + 4)(x − 4), and 4x² − 9 = (2x + 3)(2x − 3). Notice that there is no x term in these expressions; the key is that both terms are perfect squares separated by a minus sign.

平方差是 IGCSE 考卷中非常常见的一种特殊模式：a² − b² = (a + b)(a − b)。例如，x² − 16 = (x + 4)(x − 4)，4x² − 9 = (2x + 3)(2x − 3)。注意这些表达式中没有 x 项；关键在于两项都是完全平方且中间是减号。

The perfect square pattern is equally important: (x + p)² = x² + 2px + p² and (x − p)² = x² − 2px + p². For example, x² + 10x + 25 = (x + 5)² because 2 × 5 = 10 and 5² = 25. Recognising these patterns immediately saves valuable time in an examination.

完全平方模式同样重要：(x + p)² = x² + 2px + p² 和 (x − p)² = x² − 2px + p²。例如，x² + 10x + 25 = (x + 5)²，因为 2 × 5 = 10 且 5² = 25。立即识别这些模式能在考试中节省宝贵时间。

4. Solving by Factorisation | 因式分解法求解

When a quadratic can be factorised, this is usually the fastest and most reliable method. The underlying rule is the zero product property: if the product of two expressions is zero, then at least one of the two expressions must be zero. Consider x² − 5x + 6 = 0.

当二次方程可以因式分解时，这通常是最快也是最可靠的方法。其背后的规则是零积性质：如果两个表达式的乘积为零，那么两个表达式中至少有一个为零。考虑 x² − 5x + 6 = 0。

x² − 5x + 6 = (x − 2)(x − 3) = 0

Hence x − 2 = 0 or x − 3 = 0, which gives x =

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）