Algebra and graphs
IGCSE Mathematics Topic 2 19:38 English narration · English + 中文 subtitles burned in
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Transcript
Look at the main cable of this bridge.
看这座桥的主缆。
That curve is not an accident.
这条曲线不是偶然的。
Every cable like it hangs in the same shape, and one short rule describes every point on it.
每一根这样的缆索都挂成同样的形状， 而一条简短的规则就能描述它上面的每一个点。
That rule is algebra: letters standing for numbers, so one line answers a thousand questions.
这条规则就是代数：用字母代表数字， 所以一行式子就能回答一千个问题。
Today we turn that language into pictures.
今天，我们把这门语言变成图像。
Here is the plan.
这是今天的计划。
First we tidy algebra up: collect like terms, expand brackets, factorise.
先把代数整理好：合并同类项、展开括号、因式分解。
Then we solve — equations, simultaneous equations, inequalities.
然后解题：解方程、解联立方程、解不等式。
Then sequences.
接着是数列。
Then graphs: the shapes you must know, and the graphs of real journeys.
再看图像：必须认识的基本形状，以及真实行程的图像。
We finish with quadratics, solved twice.
最后我们把二次方程解两遍。
In algebra, letters stand for numbers.
在代数里，字母代表数字。
A letter whose value can change is a variable.
值可以改变的字母叫做变量。
To substitute means to put a number in place of a letter.
代入就是把数字放进字母的位置。
Find the value of three x squared minus two y, when x is four and y is five.
求三x的平方减二y的值，当x等于四、y等于五时。
First square the four: that is sixteen.
先把四平方：得到十六。
Three times sixteen is forty-eight.
三乘十六是四十八。
Two times five is ten.
二乘五是十。
Forty-eight take away ten leaves thirty-eight.
四十八减十，剩下三十八。
A term is one piece of an expression.
项是表达式中的一部分。
Like terms have exactly the same letters, raised to the same powers, and only like terms can be added.
同类项的字母完全相同，次数也完全相同，只有同类项才能相加。
Here the blue terms match, so two of them plus five makes seven.
这里蓝色的项是同类项，两个加五个就是七个。
The orange terms match, so three take away nine leaves minus six.
橙色的项也是同类项，三个减九个就剩负六个。
The plain numbers give three.
纯数字合起来是三。
The letters never change — only the coefficient in front.
字母始终不变，变的只是前面的系数。
Expanding can be seen as an area.
展开可以看成面积。
A rectangle three units tall and x plus two units wide splits into two parts: three by x, and three by two.
一个高三、宽为x加二的长方形，可以拆成两块：三乘x，以及三乘二。
So three times the bracket x plus two becomes three x plus six.
所以三乘以括号x加二，就变成三x加六。
Multiply every term inside by the term outside — that is distributing.
把括号外的项乘进括号里的每一项——这就是分配律。
The area model makes the rule hard to forget.
面积模型让这条规则很难忘掉。
Expanding removes the brackets.
展开就是去掉括号。
Multiply every term inside by the term outside.
把括号外面的项，乘进括号里的每一项。
Draw a small grid: one bracket down the side, the other along the top.
画一个小格子：一个括号放在左边，另一个放在上边。
Fill each box by multiplying its row by its column.
每个格子用它所在的行乘以列填出来。
Then collect the like terms in the middle: minus eight plus one leaves minus seven.
然后合并中间的同类项：负八加一，剩下负七。
So the answer is two x squared, minus seven x, minus four.
所以答案是二x的平方，减七x，减四。
Watch the signs: a minus belongs to the term after it, and travels with it into the boxes.
注意符号：负号属于它后面那一项，会跟着它一起进入格子。
For three brackets — Extended work — expand two first, then multiply by the third.
三个括号——这是拓展内容——先展开两个，再乘第三个。
Take x minus two, times x plus three, times two x plus one.
取x减二，乘以x加三，再乘以二x加一。
First pair: x minus two times x plus three.
先算前一对：x减二乘x加三。
That gives x squared plus x minus six.
得到x的平方加x减六。
Now multiply by the third: two x plus one.
再乘第三个：二x加一。
Two x cubed, plus three x squared, minus eleven x, minus six.
得到二x的三次方，加三x的平方， 减十一x，减六。
Two steps, same grid idea each time.
两步完成，每一步都用同样的格子思路。
Factorising is expanding in reverse: you put the brackets back.
因式分解是展开的逆过程：把括号还原回去。
Always take out a common factor first — here three x divides both terms.
永远先提公因式——这里三x能整除这两项。
Then learn to spot the difference of two squares: one square take away another splits into a sum times a difference.
然后要学会认出平方差：一个平方减另一个平方，可以分解成一个和乘一个差。
For a quadratic, split the middle term, then group in pairs.
至于二次式，把中间那一项拆开，再两两分组。
Common factor first, every time.
永远先提公因式。
Three patterns to lock in.
三种要牢记的模式。
Grouping four terms: take a common factor from each pair.
四项分组：从每一对里提出公因式。
x y plus two x, plus three y plus six, becomes x plus three times y plus two.
x y加二x，加三y加六，变成x加三，乘以y加二。
Next, difference of two squares: nine x squared minus sixteen is three x plus four, times three x minus four.
接着是平方差：九x的平方减十六，等于三x加四，乘以三x减四。
And a perfect square: x squared plus six x plus nine is x plus three, all squared.
还有完全平方：x的平方加六x加九，就是x加三的平方。
Spot the pattern, write the product.
认出模式，写出乘积。
Completing the square rewrites x squared plus b x plus c as a bracket squared, plus or minus a number.
配方法把x的平方加b x加c，改写成一个括号的平方，再加或减一个数。
Take half of the x-coefficient, square it, then balance.
取x系数的一半，平方，再平衡。
For x squared plus six x plus one: half of six is three, and three squared is nine.
对x的平方加六x加一：六的一半是三，三的平方是九。
So it becomes x plus three, all squared, take away eight.
所以变成x加三的平方，再减八。
That is completed-square form.
这就是配方形式。
When the x squared has a coefficient, take it out of the first two terms first.
当x的平方前面有系数时，先从前面两项里提出来。
Two x squared plus eight x plus three becomes two times a bracket squared, minus five.
二x的平方加八x加三，变成二乘一个括号的平方，再减五。
An algebraic fraction has algebra on the top or bottom.
代数分式就是分子或分母里带有代数。
Add and subtract using a common denominator.
加减时要用公分母。
x over three, plus x minus four over two: write both over six, then simplify to five x minus twelve, over six.
x除以三，加x减四除以二：都写成六做分母，再化简成五x减十二，除以六。
Multiply and divide as with ordinary fractions — flip the second when you divide.
乘除和普通分数一样——除法时把第二个反过来乘。
To simplify a rational expression, factorise the top and bottom, then cancel common brackets.
化简有理式时，先分解分子和分母，再约去相同的括号。
x squared minus two x, over x squared minus five x plus six, cancels to x over x minus three.
x的平方减二x，除以x的平方减五x加六，约简后是x除以x减三。
The laws of indices work with letters too.
指数法则对字母同样成立。
Same base: multiply means add the powers, divide means subtract, and a power of a power multiplies.
同底数：乘就加指数，除就减指数，幂的幂就乘指数。
Five x cubed, all squared, is twenty-five x to the six.
五x的三次方，整体再平方，是二十五x的六次方。
Twelve a to the five, divided by three a to the minus two, is four a to the seven — subtracting a negative adds.
十二a的五次方，除以三a的负二次方，是四a的七次方——减去一个负数等于加上。
You can also solve simple index equations: write both sides with the same base.
解简单的指数方程时：把两边写成相同的底数。
Two to the power x equals thirty-two, and thirty-two is two to the five, so x equals five.
二的x次方等于三十二，而三十二是二的五次方，所以x等于五。
An equation is a balance: do the same thing to both sides and it stays true.
方程就是一架天平：对两边做同样的事，等式就一直成立。
Start by expanding the bracket on the right.
先把右边的括号展开。
Now gather the letters on one side and the numbers on the other — add two x to both sides, then take twenty-one from both.
然后把字母移到一边，数字移到另一边——两边同时加二x， 再两边同时减二十一。
That leaves minus sixteen equals five x.
剩下负十六等于五x。
Finally divide both sides by five.
最后两边同时除以五。
So x is minus sixteen over five.
所以x等于负十六除以五。
Then put your answer back and check.
最后把答案代回去检验一下。
A fractional equation has the unknown in a denominator — Extended work.
分式方程的未知数在分母里——这是拓展内容。
Multiply both sides by the denominator to clear it.
两边同时乘以分母，就能去掉分母。
Here x over two x plus one equals four.
这里x除以二x加一等于四。
Multiply both sides by two x plus one: x equals four times that bracket, which is eight x plus four.
两边乘以二x加一：x等于四倍那个括号，就是八x加四。
Subtract eight x: minus seven x equals four.
减去八x：负七x等于四。
So x is minus four over seven.
所以x等于负四除以七。
Always check the answer does not make a denominator zero.
一定要检查答案会不会让分母变成零。
Two equations, two unknowns.
两个方程，两个未知数。
Each line here is every point that fits one of them, and a line has endless points — so one equation alone is never enough.
这里的每一条直线，都是满足其中一个方程的所有点； 而一条直线上有无数个点——所以只有一个方程永远不够。
Only one point sits on both lines at once.
只有一个点同时落在两条直线上。
Watch the two dots travel until they meet: that is the answer, x equals two, y equals three.
看这两个点移动，直到相遇：那就是答案，x等于二，y等于三。
In the exam, add or subtract the equations to remove a letter.
考试时，把两个方程相加或相减，消去一个字母。
For one linear and one quadratic equation — Extended — substitute the linear into the curve.
一个一次、一个二次方程——拓展内容——把一次式代入曲线。
Solve y equals x plus two, and y equals x squared.
解y等于x加二，以及y等于x的平方。
Replace y: x squared equals x plus two.
用x加二替换y：x的平方等于x加二。
Bring all to one side: x squared minus x minus two equals zero.
移到一边：x的平方减x减二等于零。
Factorise: x minus two, times x plus one.
因式分解：x减二，乘以x加一。
So x is two, giving y equals four, or x is minus one, giving y equals one.
所以x等于二，此时y等于四；或者x等于负一，此时y等于一。
Two meeting points this time.
这次有两个交点。
To change the subject of a formula means to rearrange it so a chosen letter is alone on one side.
公式变形，就是把式子重新整理，让选定的字母单独在一边。
Make r the subject of A equals pi r squared.
把A等于圆周率乘r的平方中的r变成主体。
Divide both sides by pi, then take the square root: r is the square root of A over pi.
两边除以圆周率，再开平方： r等于A除以圆周率的平方根。
When the letter appears twice — Extended — collect those terms and factorise.
当字母出现两次时——拓展内容——合并那些项再因式分解。
To make x the subject of y equals x plus one over x minus one: multiply both sides by the denominator, gather the x terms, factorise out the x, then divide.
把y等于x加一除以x减一中的x变成主体：两边乘分母，合并含x的项，提出x，再除。
So x is one plus y, over y minus one.
所以x等于一加y，除以y减一。
An inequality says one side is smaller or larger.
不等式说的是一边比另一边小或大。
Solve it like an equation, with one warning: multiply or divide by a negative and the sign turns around.
解法和方程一样，只有一个警告： 乘以或除以一个负数，符号就要反过来。
Here three x minus two is trapped between minus three and seven.
这里三x减二被夹在负三和七之间。
Add two to every part, then divide every part by three: x runs from minus one third up to three.
每一部分都加二，再每一部分都除以三：x从负三分之一到三。
A closed circle includes the end value; an open circle does not.
实心圆表示端点包含在内，空心圆表示不包含。
On a number line, draw what the inequality means.
在数轴上画出不等式的含义。
A closed circle at minus one third shows that end is included, because of the less-than-or-equal sign.
负三分之一处的实心圆表示该端点包含在内， 因为用的是小于等于号。
An open circle at three shows three is not included.
三处的空心圆表示三不包含。
Join them with a thick segment: every point on that segment is a solution.
用粗线段连起来： 线段上每一点都是解。
Sketch it every time — examiners award the diagram.
每次都画出来——阅卷老师会给图示分。
An inequality in two letters describes a region of the graph — Extended.
含两个字母的不等式描述的是图像上的一个区域——拓展内容。
Draw the boundary line first: use a broken line for strict inequalities, less than or greater than, and a solid line when equals is allowed.
先画边界线：严格不等用虚线，小于或大于；允许等于时用实线。
Then shade the unwanted side, or list the inequalities that define a given region.
然后把不要的一侧涂掉，或者列出定义某区域的不等式。
Test a point: if the origin is easy, plug in zero zero and see whether the inequality holds.
用一个点检验：原点方便时，代入零零，看不等式是否成立。
To find the nth-term rule, watch how the terms change.
要找出通项规律，就看各项是怎么变化的。
Here the terms go up by three every time, so the rule starts with three n.
这里每一项都比前一项多三， 所以规律里先写三n。
Three ones is three, but the first term is two — so subtract one.
三个一是三，但第一项是二——所以要减一。
The rule is three n minus one.
规律就是三n减一。
Below, the gaps are not equal: three, five, seven.
下面这一列，间隔并不相等：三、五、七。
So look at the gaps between the gaps — all two.
那就看间隔之间的间隔——都是二。
A constant second difference means the rule has n squared.
二阶差是常数，说明规律里含有n的平方。
Take n squared off each term and one is left.
把每一项减去n的平方，都剩下一。
So the rule is n squared plus one.
所以规律是n的平方加一。
Nature builds sequences too — this broccoli grows in self-similar spirals.
自然界也会构造数列——这棵花椰菜以自相似的螺旋生长。
A cubic sequence such as one, eight, twenty-seven, sixty-four has nth term n cubed.
三次数列，比如一、八、二十七、六十四，通项是n的三次方。
An exponential sequence multiplies by a fixed number each time: two, six, eighteen, fifty-four multiplies by three, so the rule is two times three to the power n minus one.
指数数列每次乘以固定的数：二、六、十八、五十四，每次乘三， 所以规律是二乘三的n减一次方。
Term-to-term rules tell you the next term; the nth-term rule jumps straight to any position.
递推规则告诉你下一项；通项规则可以直接跳到任意位置。
Two quantities are in direct proportion if one is always a fixed multiple of the other: y is proportional to x means y equals k x, where k is a constant.
两个量成正比例，是指一个始终是另一个的固定倍数：y与x成正比，意味着y等于k乘x，k是常数。
They are in inverse proportion if one rises as the other falls: y equals k over x.
成反比例，是指一个增大时另一个减小：y等于k除以x。
You can also have proportion to a square, square root, or cube.
也可以与平方、平方根或立方成比例。
Worked example: y is in direct proportion to x, and y is twelve when x is three.
例题：y与x成正比例，且x等于三时y等于十二。
First find k: twelve equals k times three, so k is four.
先求k：十二等于k乘三，所以k等于四。
Then when x is seven, y is twenty-eight.
当x等于七时，y等于二十八。
Always find k first from the given pair.
永远先从给定的一对值求出k。
Now a real journey.
现在看一段真实的行程。
On a distance-time graph, time runs along the bottom and distance from home goes up the side.
在距离—时间图中，时间沿着横轴，离家的距离沿着纵轴。
The steepness, the gradient, is the speed.
图线的陡峭程度，也就是斜率，表示速度。
The first part climbs steadily, so the speed is steady.
第一段匀速上升，说明速度恒定。
The middle part is flat: distance is not changing, so the object has stopped.
中间一段是水平的：距离没有变化，说明物体停下来了。
The last part drops back to zero — travelling home.
最后一段回落到零——正在返回家里。
Steeper means faster.
越陡就越快。
A conversion graph is a straight line used to change between two units — for example, miles and kilometres.
换算图是一条直线，用来在两种单位之间换算——比如英里和千米。
The gradient is the conversion factor.
斜率就是换算因子。
To convert, read across and up from one axis to the other.
换算时，从一条轴读到另一条轴。
It is the same skill as any linear graph: gradient is a rate of change.
这和任何一次函数图像的技能相同：斜率是变化率。
Practical graphs always mean something real on both axes.
实用图像的两条轴上，永远对应着真实的量。
Now put speed up the side instead.
现在把纵轴换成速度。
The gradient is now the acceleration — or deceleration if the speed falls: this journey gains three metres per second, every second.
斜率现在表示加速度——如果速度下降就是减速度：这段行程 每一秒增加三米每秒。
The area underneath is the distance travelled.
图线下方的面积，就是走过的距离。
Cut it into easy shapes — a triangle, a rectangle, another triangle: twenty-four, forty-eight, twenty-four. Ninety-six metres in total.
把它切成简单的图形—— 一个三角形、一个长方形、再一个三角形：二十四、四十八、二十四，一共九十六米。
Gradient for acceleration, area for distance.
斜率对应加速度，面积对应距离。
Know these six shapes.
先记住这六种形状。
A linear equation gives a straight line.
一次方程给出一条直线。
A quadratic gives a parabola, a smooth U — upside down if the squared term is negative.
二次方程给出抛物线， 一条平滑的U形——如果平方项是负的，就倒过来。
A cubic gives an S-shaped curve.
三次方程给出S形曲线。
A reciprocal gives two branches that near the axes but never touch.
反比例给出两条分开的曲线，靠近坐标轴却永远不碰到。
An exponential grows faster and faster, or fades towards a flat line.
指数函数要么越涨越快，要么衰减趋近一条水平线。
Recognise the shape and half the question is done.
认出形状，题目就做完一半。
A quick sketch should show the right shape and the key features.
快速草图要画出正确的形状和关键特征。
Make a table of values: choose x, work out y, plot the coordinates, join with a smooth curve.
先做数值表：选定x，算出y，描点，用平滑曲线连接。
The points where a graph crosses the x-axis are the roots — the solutions of y equals zero.
图像与x轴相交的点就是根——也就是y等于零的解。
You can solve by reading a graph: the intersection point of a line and a curve gives the joint solution.
也可以从图上读解：直线与曲线的 交点就是两个方程的联立解。
An asymptote is a line the curve gets closer and closer to but never touches — the axes for a reciprocal, or a horizontal line for exponential growth and decay.
渐近线是曲线越来越靠近却永远不碰到的直线—— 反比例靠近坐标轴，指数增长与衰减靠近一条水平线。
The y-intercept is where it crosses the vertical axis.
y截距是与纵轴的交点。
Mark symmetry when you see it.
看到对称就标出来。
Look closely at the parabola.
仔细看这条抛物线。
It has one lowest point, and it is symmetric about a vertical line through it.
它有一个最低点，并且关于经过这一点的竖直线对称。
Completing the square finds both at once.
配方法可以一次求出这两样东西。
Here the equation is a bracket squared, take away eight.
这里方程写成一个括号的平方，再减八。
A square is never negative, so the smallest value comes when the bracket is zero: x equals minus three, y equals minus eight.
平方永远不会是负数，所以括号等于零时取到最小值：x等于负三，y等于负八。
That point is the turning point, and the dashed line is the line of symmetry.
那一点就是顶点，那条虚线就是对称轴。
The points where a curve crosses the horizontal axis are the roots, and finding them means solving the quadratic.
曲线与横轴相交的那些点就是根，求根就是解这个二次方程。
Three routes.
有三条路。
Factorise, when the numbers are friendly.
第一，因式分解，数字友好时用它。
Complete the square, when the question asks for it.
第二，配方法，题目要求时用。
Or use the formula, printed in the exam, and it always works.
第三，用求根公式，考试里已经印给你了，而且永远有效。
A quadratic usually has two answers — give both.
二次方程通常有两个解——两个都要写。
Solve by completing the square.
用配方法求解。
Start with x squared plus six x plus one equals zero.
从x的平方加六x加一等于零开始。
Move the one: x squared plus six x equals minus one.
移开一：x的平方加六x等于负一。
Complete the square on the left: x plus three, all squared, equals eight.
左边配方：x加三的平方等于八。
Take the square root of both sides: x plus three equals plus or minus the square root of eight, which simplifies to plus or minus two root two.
两边开平方：x加三等于正负根号八， 也就是正负二倍根号二。
So x is minus three plus or minus two root two.
所以x等于负三加或减二倍根号二。
Leave exact answers unless asked for decimals.
除非题目要求小数，否则保留精确答案。
Solve two x squared, plus seven x, plus three, equals zero.
解：二x的平方，加七x，加三，等于零。
Pause and try it first.
先暂停，自己试一试。
Multiply the first number by the last: two times three is six.
把第一个数和最后一个数相乘：二乘三等于六。
Now find two numbers that multiply to six and add to seven: one and six.
再找两个数，乘起来是六，加起来是七：一和六。
Split the middle term into one x plus six x.
把中间那一项拆成一x加六x。
Group the four terms in pairs; each pair leaves the same bracket, so it comes out once.
把四项两两分组；每一组都剩下同一个括号，所以它可以提出来一次。
That gives x plus three, times two x plus one.
于是得到x加三，乘以二x加一。
A product is zero only when a factor is zero.
乘积等于零，只可能是某个因式等于零。
So x is minus three, or x is minus one half.
所以x等于负三，或者x等于负二分之一。
The same question, now with the formula.
同一道题，现在改用求根公式。
Write the three numbers down first: a is two, b is seven, c is three.
先把三个数写下来：a是二，b是七，c是三。
Substitute them with brackets around every value.
代入时给每个数都加上括号。
Inside the square root, seven squared is forty-nine, take away four times two times three, leaving twenty-five, whose square root is five.
根号里面：七的平方是四十九，减去四乘二乘三，剩下二十五， 它的平方根是五。
So the top is minus seven, plus or minus five, and the bottom is four.
所以分子是负七加或减五，分母是四。
Plus five gives minus one half; minus five gives minus three.
取加五得到负二分之一；取减五得到负三。
The same two answers as before.
和刚才是同样的两个答案。
Use it whenever the factors will not come.
只要分解不出来，就用它。
Differentiation finds the gradient of a curve at any point — Extended work.
微分可以求出曲线上任意一点的斜率——这是拓展内容。
You can estimate it by drawing a tangent, a line that just touches the curve, and measuring its gradient.
你可以画一条切线，也就是刚好碰到曲线的直线，再量它的斜率来估计。
The exact rule is cleaner: if y is a times x to the n, the derivative is a times n times x to the n minus one.
精确法则更干净：如果y是a乘x的n次方，那么导数就是a乘n再乘x的n减一次方。
Differentiate a sum term by term.
对和式逐项求导。
The derivative is written d y by d x.
导数写成d y除以d x。
Worked example.
例题。
If y is x cubed plus two x squared minus five x, then the derivative is three x squared plus four x minus five — differentiate a sum term by term.
如果y是x的三次方加二x的平方减五x，那么导数是三x的平方加四x减五——对和式逐项求导。
A stationary point, or turning point, is where the gradient is zero, so set the derivative equal to zero.
驻点也就是转折点，是斜率等于零的地方，所以令导数等于零。
For y equals x squared minus six x plus five: the derivative is two x minus six equals zero, so x is three.
对y等于x的平方减六x加五：导数是二x减六等于零，所以x等于三。
Then y is minus four.
然后y等于负四。
The turning point is three, minus four.
转折点是三点负四。
Check the sign of the gradient on each side to tell maximum from minimum.
看两侧斜率的正负，可以区分最大值和最小值。
A function turns each input into one output.
函数把每个输入变成一个输出。
We write f of x, for example f of x equals three x minus five, so f of two equals one.
我们写成f of x，例如f of x等于三x减五，所以f of二等于一。
The set of allowed inputs is the domain; the set of possible outputs is the range.
允许的输入集合叫定义域；可能的输出集合叫值域。
Think of a machine: times three, then minus five.
把它想成一台机器：先乘三，再减五。
The inverse machine runs it backwards — reverse the order, undo each step: plus five, then divide by three.
反函数机器倒着运行——颠倒顺序，撤销每一步：先加五，再除以三。
To find the inverse, write y equals f of x, swap the roles, and make x the subject.
求反函数时，写出y等于f of x，交换角色，再把x变成主体。
For f of x equals three x minus five, the inverse is x plus five, all over three.
对f of x等于三x减五，反函数是x加五，再整体除以三。
A composite function applies one function after another: g f of x means do f first, then g.
复合函数是一个接一个应用：g f of x表示先做f，再做g。
If f of x is two x and g of x is x plus three, then g f of x is two x plus three, but f g of x is two x plus six.
如果f of x是二x，g of x是x加三，那么g f of x是二x加三，但f g of x是二x加六。
Order matters — g f is not the same as f g.
顺序很重要——g f和f g并不相同。
Three marks students throw away, and one habit that saves them.
三个学生常丢的分，还有一个能救分的习惯。
First, a minus sign in front of a bracket changes every sign inside, not only the first.
第一，括号前面的负号会改变括号里每一项的符号， 不只是第一项。
Second, flip the inequality sign whenever you divide by a negative.
第二，只要除以一个负数，不等号的方向就要反过来。
Third, factorise fully — common factor first — and give both answers to a quadratic.
第三，要分解彻底——先提公因式——解二次方程时两个答案都要写。
And put brackets around every value you substitute.
还有，代入时给每个数都加上括号。
Get these right and this topic is yours.
把这些做对，这个专题就是你的了。