Edexcel GCSE数学 代数 因式分解 二次方程
Edexcel GCSE 数学的代数部分是整份卷子里分值最重的一块，Higher 卷的内容配比里它占到将近三分之一。它考的事情其实很窄：把式子写对、把方程解出来、把图像与数列读明白。这个板块的知识点彼此咬合，同类项与展开是解方程的前置，因式分解又是二次方程的前置，缺一块后面就会卡住。Edexcel 1MA1 的考纲把代数分成式与运算、方程与不等式、数列与图像几个部分，三张卷都会考到。这篇文章按出题顺序拆成十二个小节，每节给出考点口径、典型例题与阅卷时容易丢分的地方。读完可以先挑两个最没把握的小节练习，不必从头背到尾。
Algebra is the heaviest single strand of Edexcel GCSE Mathematics, carrying close to a third of the marks on the Higher papers. What it really tests is narrow: writing expressions correctly, solving equations, and reading graphs and sequences. The topics build on each other, so weak notation or shaky factorising later turns into lost marks on quadratics and simultaneous equations. This article works through twelve sections in exam order, with the specification wording, worked examples and the mistakes examiners see every series.
字母、项与同类项
代数的第一步是读懂写法。3x 表示 3 乘 x，x 的平方写作 x^2，(x + 4)/2 表示 x 加 4 以后再除以 2。同类项指字母部分完全相同的项，3x 与 7x 可以合并，3x 与 3x^2 不能合并。合并时只加减前面的系数，字母部分照抄，5x - 2x + 7 整理为 3x + 7。很多学生以为 3x + 4 能写成 7x，其实 4 后面没有 x，两项不是一类。负号的归属是最常见的失分点，5 - (2x - 3) 展开后是 5 - 2x + 3，不是 5 - 2x - 3。写答案时把项按次数从高到低排好，阅卷老师一眼能看到你整理到了哪一步。
Algebra on this specification starts with notation: 3x means 3 multiplied by x, x^2 is x squared, and (x + 4)/2 means x plus 4, then divided by 2. Like terms carry identical letter parts, so 3x and 7x combine while 3x and 3x^2 cannot be added at all. Only the coefficients change when you collect, which is why 5x - 2x + 7 becomes 3x + 7. The sign in front of a bracket causes most of the damage here; 5 - (2x - 3) expands to 5 - 2x + 3, and writing 5 - 2x - 3 throws away the mark.
展开括号：单括号与双括号
展开括号就是把乘法逐项做开。2(3x + 5) 得 6x + 10；(x + 3)(x - 5) 要四次相乘，逐项算出 x^2 - 5x + 3x - 15，再合并成 x^2 - 2x - 15。三个括号相乘时先把前两个展开成一个二次式，再乘第三个，比一口气乘六次要稳。常见错误是只乘括号里的第一项，3(x + 2) 被写成 3x + 2。另一个坑在 x 的平方项，x(x + 4) 里的 x 乘 x 必须写成 x^2，漏掉指数整道题的分都会掉。检查办法很直接，把 x = 1 代回原式与展开式，两边应当相等，这一步在考场上比重新算一遍快。
Expanding brackets means multiplying every term inside by what sits outside. So 2(3x + 5) gives 6x + 10, and (x + 3)(x - 5) needs four separate products before they collect into x^2 - 2x - 15. Where three brackets appear, multiply the first two, simplify, then multiply by the third. Touching only the first term inside a bracket is the recurring slip, and dropping the index on x multiplied by x is the second. Substituting x = 1 into both forms is a quick self-check that costs seconds.
因式分解：公因式与平方差
因式分解是展开的反向操作，答案写成几个括号相乘。先找公因式，6x^2 + 9x 提出 3x，得 3x(2x + 3)。二次式 x^2 + 7x + 12 要找两个数，乘积 12、和 7，即 3 与 4，写成 (x + 3)(x + 4)。平方差 x^2 - 25 直接给出 (x - 5)(x + 5)，这类式子在化简代数分式时出现得最多。学生最常把括号里的符号写反，把 (x + 3)(x + 4) 写成 (x - 3)(x - 4)。分解完把括号乘回去验算一遍，两次结果一致才停笔，中间项最容易在这里对上或对不上。
Factorising reverses expansion and delivers the answer as a product of brackets. Common factors come first, so 6x^2 + 9x becomes 3x(2x + 3); for x^2 + 7x + 12 you need two numbers with product 12 and sum 7, namely 3 and 4, giving (x + 3)(x + 4). A difference of two squares such as x^2 - 25 factors straight away to (x - 5)(x + 5), a pattern that keeps appearing in algebraic fractions. Signs inside the brackets are where marks go missing, so multiply back out before moving on.
解一元一次方程
解方程的目标是把 x 单独留在一边。5x - 7 = 3x + 9 先把 x 集中到左边，2x = 16，x = 8。含分数系数的方程两边同乘分母，(x + 1)/3 = 4 两边乘 3 得 x + 1 = 12。括号里有减号时先展开再移项，x 的系数为负时两边同时除以负数，符号跟着翻。很多学生以为检验这一步多余，其实把答案代回原方程花不了十秒，却能挡住大部分计算失误。方程有分母时，乘的时候每一项都要乘到，漏掉常数项是最常见的失分点。移项后把同类项一次合完再动下一步，中间少写一行也比反复改符号强。
Solving a linear equation is about isolating x. With 5x - 7 = 3x + 9 you collect the x terms to get 2x = 16, so x = 8; when fractions appear, multiply every term on both sides by the denominator, which turns (x + 1)/3 = 4 into x + 1 = 12. Negative coefficients flip the signs on both sides when you divide. Substituting the answer back takes ten seconds and catches nearly all arithmetic slips, and that check is worth more when later parts of the question depend on the value you found.
不等式与数轴表示
不等式与方程的解法几乎一样，差别在最后一步：两边同乘或同除以负数时，不等号方向要反过来。-2x 大于 8 得到 x 小于 -4。答案画在数轴上，空心圈表示不包含端点，实心圈表示包含端点，x ≥ 3 是实心圈向左或向右的第一个点。同样地，x 大于 2 且小于等于 7 表示两个条件同时成立。写答案用不等式或区间都算对，方向与端点必须标清。常见的错误是把 -2x 大于 8 直接写成 x 大于 -4，丢掉的正是符号翻转这一步，而这一步常常单独占一分。
Inequalities follow the same steps as equations with one extra rule attached. Multiplying or dividing by a negative number reverses the direction of the sign, so -2x greater than 8 gives x less than -4. On a number line an open circle excludes the endpoint and a filled circle includes it, which is how x ≥ 3 is drawn. Where two inequalities are combined, the answer is the overlap of the two solution sets. Reversing the sign is the step candidates forget most often, and it usually carries a mark of its own.
代入数值与公式变形
代入就是给字母换上数字，负数的平方要先加括号，x = -3 时 x^2 = 9 而不是 -9。公式变形是把另一个字母变到等号左边，A = πr^2 解出 r 要两步，先除以 π 再开平方，得 r = sqrt(A/π)。从 v = u + at 解出 t 得 t = (v - u)/a，除过去时整个分子都在分数线上面。Edexcel 1MA1 的考纲把这一条写得很直白：substitute numerical values into formulae，change the subject of a formula。学生最常把 A = πr^2 写成 r = A/π 就停手，忘记还要开平方。变完形代一组数字进去试算，漏掉哪一步立刻显形。
Substitution means swapping letters for numbers, and a negative value such as x = -3 must be bracketed before squaring, which is why x^2 = 9 there and not -9. Rearranging moves a chosen letter to the left side: from A = πr^2 you divide by π and then take the square root, giving r = sqrt(A/π). The same plan turns v = u + at into t = (v - u)/a, with the whole numerator above the division line. The Pearson Edexcel 1MA1 specification states this requirement in exactly those words. Testing a rearranged formula with one set of numbers exposes missed steps immediately.
联立方程：消元与代入
两个未知数需要两个方程。消元法先让某个字母的系数相等，3x + 2y = 16 与 5x - 2y = 8 相加，y 直接消掉，8x = 24，x = 3，代回得 y = 3.5。系数对不上时把整个方程乘一个数再加减，注意常数项也要一起乘。代入法适合其中一个方程已经写成 y = ... 的情形，把整段式子代进另一个方程。得出了一个字母的值，一定代回原方程而不是代回自己刚写的那一行。阅卷时两个答案都要写清楚，只写 x 的值得不到后一半的分。
Two unknowns need two equations. Elimination matches the coefficients of one letter first: adding 3x + 2y = 16 and 5x - 2y = 8 removes y and leaves 8x = 24, so x = 3 and y = 3.5. Where the coefficients differ, multiply a whole equation, constant term included. Substitution suits the case where one equation already begins y = ... . Put the first value back into an original equation rather than into your own working line, and write both answers out.
二次方程：分解、配方与求根公式
二次方程 ax^2 + bx + c = 0 有三条路。能分解的优先分解，x^2 - 5x + 6 = 0 得 (x - 2)(x - 3) = 0，两个根是 2 与 3。分解不出来时用求根公式 x = (-b ± sqrt(b^2 - 4ac))/(2a)，Higher 卷的公式页上有它。配方法把式子写成 (x + p)^2 + q 的形式，用于求顶点或做证明。判别式 b^2 - 4ac 决定根的个数，正数两个根，等于零一个重根，负数没有实根，这一步能省去无效计算。题目写 exact value 时答案要给根号或分数形式，其余情况按 Pearson 的要求保留 3 位有效数字。
A quadratic ax^2 + bx + c = 0 offers three routes. Factorising is fastest when it works: x^2 - 5x + 6 = 0 becomes (x - 2)(x - 3) = 0 with roots 2 and 3. Otherwise reach for x = (-b ± sqrt(b^2 - 4ac))/(2a), which appears on the Higher formula sheet. The discriminant b^2 - 4ac tells you whether to expect two roots, one repeated root or none, saving pointless work; completing the square rewrites the expression as (x + p)^2 + q, the form you need for turning points and proofs, and non-exact answers stay at 3 significant figures.
直线与二次函数的图像
直线方程 y = mx + c 里 m 是梯度、c 是 y 轴截距，画图找两点连起来就够。平行线梯度相同，垂直线的梯度乘积为 -1，y = 2x + 1 与 y = -0.5x + 4 互相垂直。二次函数的图像是抛物线，对称轴在 x = -b/(2a)，顶点落在这条线上，把对称轴代回函数就得到顶点坐标。图像与 x 轴的交点就是方程的解，读图得到的根通常是近似值，题目要精确答案时得回到代数方法。很多学生以为画图题只考画图，其实读懂截距、交点与交点个数才是拿分的地方。
For a straight line y = mx + c, m is the gradient and c the y-intercept, and two points are enough to draw it. Parallel lines share a gradient, while perpendicular gradients multiply to -1, so y = 2x + 1 meets y = -0.5x + 4 at a right angle. A quadratic graph is a parabola whose axis of symmetry sits at x = -b/(2a), with the turning point on that line. Roots read off a graph are estimates, so switch to algebra whenever an exact value is required.
数列：通项与二阶差
数列的通项用 n 表示第几项。等差数列 5, 9, 13, 17 每次加 4，通项是 4n + 1，把 n = 1 代进去得 5，与首项对上说明形式写对了。二次数列的一阶差是等差数列、二阶差为常数，2, 7, 16, 29 的二阶差是 4，通项里含 2n^2。判断某个数是否属于这个数列，把它代入通项解方程，解出的 n 必须是正整数。常见的错误是 n 从 0 开始代入，整条式子整体错位一项。写完通项把 n = 1 与 n = 2 都验一遍，两分钟能省下一次返工。
A sequence formula uses n for the position of a term. Linear sequences such as 5, 9, 13, 17 rise by 4 each step, so the nth term is 4n + 1. Quadratic sequences show a constant second difference: 2, 7, 16, 29 has second difference 4, which points to a 2n^2 term. To test whether a number belongs to a sequence, substitute it and solve for n, which has to come out as a positive integer; checking n = 1 and n = 2 against the original list takes a minute and prevents off-by-one answers.
代数证明与恒等式
Higher 卷要求用代数做证明。偶数写成 2n，奇数写成 2n + 1，三个连续整数写成 n, n + 1, n + 2，这些是证明题的起手式。证明恒等式时从复杂的一边化到简单的一边，每一步都要能说出依据，不能两边同时改。反例只需要一个，要说某个说法不成立，找一个数字代进去即可，x 取 0.5 时 x^2 大于 x 这个说法就不成立了。阅卷时论证的连接词同样给分，所以写清每一步做了哪件事。恒等号用三条横线表示，与方程的等号含义不同，混用会被扣分。
Higher tier asks for algebraic proof. Even numbers are written 2n, odd numbers 2n + 1, and three consecutive integers become n, n + 1 and n + 2. When proving an identity, work from the messier side towards the simpler one and show what each line does. A single counterexample settles a false claim, such as x = 0.5 defeating the statement that x^2 is always greater than x. Marks are awarded for the logic between the lines, so write the words as well as the symbols.
Foundation 与 Higher 的代数差异
Pearson Edexcel 1MA1 的考纲给出内容配比范围，代数在 Foundation 约占 17-23%，在 Higher 约占 27-33%，Higher 题型多出配方法、代数证明、迭代求近似解与分式方程。三张卷各 1 小时 30 分钟、各 80 分，Paper 1 不允许使用计算器，平方与分数运算要能手算。Pearson 官网 2025 年考试窗口的公式页说明两个等级都随卷提供公式页，覆盖周长、面积、体积、圆、勾股定理与三角函数比值，扇形面积与圆锥表面积不在页上，仍要自己记。非精确答案默认保留 3 位有效数字，题目要求 exact 时给精确形式。答案写清步骤，过程分常常比最终答案多。把错题按知识点归档，比反复刷整套卷更省时间。
Algebra carries roughly 17-23% of the marks at Foundation and 27-33% at Higher, according to the Pearson Edexcel 1MA1 specification. Higher adds completing the square, algebraic proof, iteration and algebraic fractions on top of the Foundation content. Three papers, each 1 hour 30 minutes and 80 marks, with paper 1 taken without a calculator, so mental work on squares and fractions still matters. The formula sheet issued for the 2025 exam window covers perimeter, area, volume, circles, Pythagoras and the trigonometric ratios, but sector area and cone surface area stay on the memorise list. Method marks often outweigh the final number, so keep the working visible.
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