📚 Algebra 6: Manipulating and Solving Quadratics | 代数6：二次表达式的变形与求解
Welcome to this revision guide for Algebra 6, part of the Edexcel IGCSE Mathematics course. In this chapter, we focus on quadratic expressions and equations: how to expand, factorise, solve and rearrange them. These skills are essential for many later topics, including graphs, inequalities and calculus.
欢迎阅读 Edexcel IGCSE 数学课程中 Algebra 6（代数 6）的复习指南。本章我们聚焦于二次表达式与二次方程：如何展开、因式分解、求解和变形。这些技能是后续学习图象、不等式和微积分等主题的重要基础。
1. Expanding and Simplifying | 展开与化简
To expand a pair of linear brackets, multiply each term in the first bracket by every term in the second bracket. This is often remembered by the word FOIL: First, Outer, Inner, Last.
要展开两个线性括号，需将第一个括号中的每一项乘以第二个括号中的每一项。常用口诀 FOIL 记忆：First（首项）、Outer（外项）、Inner（内项）、Last（末项）。
For example, expand (2x + 3)(x – 5).
例如，展开 (2x + 3)(x – 5)。
(2x + 3)(x – 5) = 2x² – 10x + 3x – 15 = 2x² – 7x – 15
Always collect like terms after expanding. Check that the coefficient of x² is the product of the two first terms, and the constant term is the product of the two last terms.
展开后务必要合并同类项。检验时注意：x² 的系数是两个首项的乘积，常数项是两个末项的乘积。
2. Factorising Quadratic Expressions | 二次表达式的因式分解
Factorising is the reverse of expanding. For a quadratic of the form x² + bx + c, find two numbers that multiply to give c and add to give b.
因式分解是展开的逆运算。对于形如 x² + bx + c 的二次式，找到两个数，使得它们的乘积为 c，和为 b。
Factorise x² – 5x + 6. The two numbers are -2 and -3, because (-2) × (-3) = 6 and (-2) + (-3) = -5.
因式分解 x² – 5x + 6。两个数为 -2 和 -3，因为 (-2) × (-3) = 6，且 (-2) + (-3) = -5。
x² – 5x + 6 = (x – 2)(x – 3)
For a quadratic with a leading coefficient not equal to 1, such as 2x² + 7x + 3, we can use trial-and-error or the grouping method. Here the factors are (2x + 1)(x + 3).
对于首项系数不为 1 的二次式，例如 2x² + 7x + 3，可以采用试凑法或分组法。这里的因式为 (2x + 1)(x + 3)。
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: a² – b² = (a – b)(a + b). This is useful when both terms are perfect squares.
平方差是一个特殊模式：a² – b² = (a – b)(a + b)。当两项都是完全平方数时，这个公式非常有用。
Factorise 4x² – 9. Notice 4x² = (2x)² and 9 = 3², so:
因式分解 4x² – 9。注意 4x² = (2x)²，9 = 3²，所以：
4x² – 9 = (2x – 3)(2x + 3)
A perfect square trinomial follows a² + 2ab + b² = (a + b)² or a² – 2ab + b² = (a – b)². For example, x² + 6x + 9 = (x + 3)².
完全平方三项式符合 a² + 2ab + b² = (a + b)² 或 a² – 2ab + b² = (a – b)²。例如，x² + 6x + 9 = (x + 3)²。
4. Solving Quadratic Equations by Factorising | 用因式分解解二次方程
A quadratic equation in the form ax² + bx + c = 0 can be solved by factorising the left-hand side and then using the zero product property: if p × q = 0, then p = 0 or q = 0.
形如 ax² + bx + c = 0 的二次方程，可以通过对左侧因式分解，然后利用零乘积性质求解：若 p × q = 0，则 p = 0 或 q = 0。
Solve x² – 5x + 6 = 0. First factorise: (x – 2)(x – 3) = 0. Then set each factor equal to zero.
解方程 x² – 5x + 6 = 0。先因式分解：(x – 2)(x – 3) = 0。然后令每个因式等于零。
x – 2 = 0 ⇒ x = 2 or x – 3 = 0 ⇒ x = 3
Always expand your factors to check that you have the correct product. If the equation is not written as zero, rearrange it first.
始终检查你的因式乘积是否正确。如果方程未写成等于零的形式，请先移项整理。
5. The Quadratic Formula | 二次公式
When a quadratic cannot be factorised easily, use the quadratic formula. For ax² + bx + c = 0, the solutions are given by:
当二次式不易因式分解时，可使用二次公式。对于 ax² + bx + c = 0，解为：
x = ( -b ± √(b² – 4ac) ) / (2a)
The expression Δ = b² – 4ac is called the discriminant. If Δ > 0, there are two distinct real roots; if Δ = 0, one repeated root; if Δ < 0, no real roots.
表达式 Δ = b² – 4ac 称为判别式。若 Δ > 0，有两个不同实根；若 Δ = 0，有一个重根；若 Δ < 0，没有实根。
Solve 2x² + 3x – 5 = 0 using the formula. Here a = 2, b = 3, c = -5.
用公式解 2x² + 3x – 5 = 0。这里 a = 2，b = 3，c = -5。
x = ( -3 ± √(3² – 4 × 2 × (-5)) ) / (2 × 2) = ( -3 ± √49 ) / 4 = ( -3 ± 7 ) / 4
Therefore x = 1 or x = -5/2. Always simplify your answers as much as possible.
因此 x = 1 或 x = -5/2。答案要尽可能化简。
6. Completing the Square | 配方法
Completing the square rewrites a quadratic in the form (x + p)² + q. Start with x² + bx: take half of b, write (x + b/2)², then subtract (b/2)².
配方法将二次式改写为 (x + p)² + q 的形式。从 x² + bx 开始：取 b 的一半，写成 (x + b/2)²，然后减去 (b/2)²。
Rewrite x² + 6x + 1 in completed square form. Half of 6 is 3, so add and subtract 3².
将 x² + 6x + 1 写成配方形式。6 的一半是 3，因此加上并减去 3²。
x² + 6x + 1 = (x + 3)² – 9 + 1 = (x + 3)² – 8
Completing the square is also useful for finding the turning point of a quadratic graph: the vertex is (-p, q) for y = (x + p)² + q.
配方法也常用于寻找二次图象的顶点：对于 y = (x + p)² + q，顶点坐标为 (-p, q)。
7. Algebraic Fractions | 代数分式
Algebraic fractions are simplified by factorising the numerator and denominator, then cancelling common factors. Remember that division by zero is not allowed, so state any restrictions on the variable.
代数分式通过因式分解分子分母、消去公因式来化简。注意分母不能为零，因此需注明变量的限制条件。
Simplify (x² – 1) / (x – 1). Factorise the numerator as a difference of two squares.
化简 (x² – 1) / (x – 1)。将分子按平方差公式分解。
(x² – 1) / (x – 1) = (x – 1)(x + 1) / (x – 1) = x + 1 (x ≠ 1)
To add or subtract algebraic fractions, first find a common denominator, just as with numerical fractions.
加减代数分式时，与数字分式一样，需要先找到公分分母。
8. Formula Rearrangement | 公式变形
Changing the subject of a formula may require solving a quadratic. Rearrange all terms to one side, then solve for the required variable.
改变公式的主项时，可能会需要解二次方程。将所有项移到一边，然后求解所需变量。
Make x the subject of 5x² + 2 = 17. Start by isolating x².
在 5x² + 2 = 17 中将 x 表示出来。首先分离 x²。
5x² = 15 → x² = 3 → x = ±√3
When taking the square root of both sides, always include the ± sign unless the context only allows a positive value, such as a length or time.
对方程两边开平方根时，除非题意只允许正值（如长度或时间），否则必须加上 ± 号。
In this guide, we have covered the key techniques of Algebra 6: expanding, factorising, solving quadratics by factorisation, using the quadratic formula, completing the square, simplifying algebraic fractions and rearranging formulas. Practise these skills regularly, and always check your work by substituting your answers back into the original equation.
在本指南中，我们复习了 Algebra 6 的关键技巧：展开、因式分解、用因式分解解二次方程、使用二次公式、配方法、化简代数分式以及公式变形。请经常练习这些技能，并始终将答案代入原方程进行检验。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply