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如何简单破解一元二次方程式

How to Factor any Quadratic Equation Easily - Trick for factorising

大家好 欢迎收看《数学技巧》
G’day, welcome to Tacmath Channel.
今天我们的节目是讲
What we’re gonna be having a look at today’s video
如何很容易地解一元二次方程
is how to factorize quadratic equations really simply, really easily.
首先 我来举个例子
So I’ll put an example up for you.
咱们要解的二元一次方程是
So the equation we’re going to have a look at is this one,
2x²+7x-4=0
2x squared plus 7x take away 4 is equal to 0.
这个方程怎么解?很简单
Now, how do we tackle this one? Really simply.
我们只需要用系数 也就是x²或x前面的数字
We are just gonna look at the coefficients, that is the numbers in front of our x squared or x here,
还有常数来解 这个方程里的常数是-4
as well as this our constant here, there’s negative 4, to get our answer.
好 我们先做上标记
So we’re just gonna label this.
A是2 B是7
We got A which is a 2, we got B which is a 7
C是-4
and C which is a negative 4.
这方法适用于所有这种类型的方程
It’s gonna be the same pattern again and again, whatever you solve any of these.
好 接下来我们要做的第一件事是
Ok, so the first thing we do after we do that is
把A和C相乘 即第一个数乘以最后一个数
we just go A times C, this first number times the last number.
也就是2乘以-4
2 times negative 4,
得到的结果是-8
2 times negative 4 is equal to negative 8.
然后我们要到用中间这个系数 也就是B
We’re just gonna use this middle coefficient here for our next part which is B.
B在这里是7
B is equal to 7.
接下来呢?我们需要两个数字 -8的两个因数
Now what? Well, we’re gonna look for two numbers – two factors of negative 8.
如果这两个数相乘等于-8
Then when you multiply, you have negative 8,
那它们就是-8的因数
they would make the factors of negative 8,
但同时把这两个数相加 会得到7
but when you add them, give us 7.
所以这两个数字必须 相乘得-8 相加得7
So two numbers you multiply to give negative 8 but add to give 7 here.
同时满足这两点的
The factors of negative 8 that add to give 7 are the following
是8和-1
we have 8 and negative 1.
因为8乘以-1等于-8
Ok, because when we multiply 8 times negative 1, we get negative 8.
而把8和-1相加
But when we get 8 and negative 1 and put them together,
即8减去1等于7
we get 8 minus 1 which is 7.
然后呢?然后就很简单了
Now what? Well, pretty simple.
我们只需要变换一下这两个数的符号
We are just going to change the signs on these.
我的意思是
What I mean by that is
所有的负号变成正号 所有的正号变成负号
all things negative will become positive and all things positive will become negative.
这样变换下来 我们就得到了-8和1
Negative 8 and positive 1, just flip those around.
最后 我们只需要用-8和1除以第一个系数A
Last of all, we’re just gonna put this over, the coefficient A here, ok, the very first one here,
也就是2 都除以2
divided by 2, divided by 2.
这样我们就得到了x的两个值
And this gives us here two values for x.
这两个值是什么呢?一个是-8除以2
What x could equal? X could be equal to negative 8 over 2,
答案是-4
so negative 8 divided by 2 is negative 4
另外一个是1除以2 即1/2
or x could be also equal to positive 1 over 2 which is a half.
很简单吧?要记住这个方式哦
Pretty simple, right? Just remember that pattern, ok?
让我们再来看下一个
Let’s have a look at another one.
接下来的一元二次方程是
The next one we’ll have is
2x²-2x-12=0
2x squared minus 2x minus 12 is equal to 0.
你自己试一下吧
You give it a fly.
好 A B C标记好了
Ok, we have A B C.
然后把A和C相乘
We’re gonna multiply A times C,
即 2乘以-12等于-24
this is equal to 2 times minus 12, which is minus 24.
B是-2
B is equal to negative 2.
哪两个数字相乘得-24
What numbers multiply to give negative 24,
相加得-2?
but then you can put together to get negative 2?
答案是4和-6
4 and minus 6.
因为4乘以-6等于-24
Again, because if you multiply 4 and minus 6, you get negative 24,
4减去6等于-2
but if you go 4 take away 6 you get negative 2.
接下来呢?
So what do we do?
我们要变换符号
We are going to flip the symbols here,
负号变正号 正号变负号
everything negative becomes positive, everything positive becomes negative.
所以我们得到了-4和6
We have negative 4 and positive 6
然后把这两个数除以系数2 还记得吧?
and we are going to put these over, you got it? this coefficient here, over 2.
这样就得到x的两个值了
And these are gonna be our two solutions.
其中一个是
X is going to be equal to, to solve this,
-4除以2等于-2
minus 4 divided by 2 which is negative 2,
另一个是6除以2等于3
or 6 divided by 2, which is 3.
你做得怎么样?我们再来一个更难的
Ok, how’d you go with that? Let’s make it even harder.
-10X²+13X+3=0
All right, minus 10x squared plus 13x, plus 3 is equal to 0.
自己试一下 我给你一点提示
Give this one a go. I’ll give you a little heads up.
这个有点难
There’s a bit of a trick with this one, ok?
你需要注意因数
Just be careful of your factors.
标好了ABC 如果你觉得自己没问题 就按“暂停”
All right, so A B C. Hit pause if you’re good at it, ok? Let’s just go
我们开始吧 A乘以C 即-10乘以3等于-30
A times C which is equal to, now minus 10 times 3, which is negative 30.
B是13
And B is equal to 13.
这一步找因数有点难
Now, well I can see there’s a bit of trick with factors here.
你是不是觉得这两个因数是10和3呢?
Do you wanna look at it and think it okay with 10 and 3 ?
因为10加3等于13
because you know 10 and 3 can be used to get 13.
但它们不是-30的因数 因为它们相乘并不是负数
But they don’t do their factors, ok?Because, it’s gonna go to a negative.
所以要么是10和-3 要么是-10和3
It’ll has to be 10, negative 3 or negative 10 and 3.
但它们相加怎么也不等于13
You can’t put those together to get 13.
实际上 正确的因数是
In fact, their factors are the following and factors are going to be
15和-2
15 and negative 2.
因为15乘以-2等于-30
Because we multiply these to get 30
15减去2等于13
and we put 15 take away 2 is 13.
所以它俩才是-30的因数
So they’re the two factors.
又到了变换符号的时候了 黑变白 白变黑
Once again, black becomes white and white becomes black,
负号变正号 白天变夜晚
negative becomes positive, day becomes night.
我们就得到了-15和2
And negative 15, positive 2.
然后用它们除以系数
And we put those over the coefficient.
这里系数是-10 都除以-10
To be honestly, what x is equal to,
这样就得到了x的两个值
the coefficient is negative 10, negative 10.
所以这里是-15除以-10
Ok, so negative 15 divided by negative 10,
两个负号可以消掉
x could be equal to,
那么x的其中一个值是
ok, that’s gonna cancel each other out,
15除以10等于1.5
15 divided by 10 is one and a half.
另外一个值是 2除以10等于1/5
2 divided by minus 10, we got a negative one-fifth.
你自己做得怎么样?
How did you go with that?
接下来的最后一题 我们来个超级难的!
I’ll tell you what, we’ll get a doozy for the last one.
这个方程是 6x²-x-2=0
Here we go, 6x squared take away x, take away 2 is equal to zero.
求解x
Solve for X a
你们应该都知道我接下来要做什么了吧
Now this is gonna be as easy as, you know where I’m going with this, A B C.
标注A B C 然后A乘以C等于-12
A times C is equal to minus 12,
B是-1
B is going to be equal to negative 1.
哪两个数相乘得-12
What two numbers multiply to give negative 12,
相加得-1呢?
but you put together you add together, you get negative 1?
我猜你的答案是不是4和3?很接近了
There you got it. I reckon you’re getting a 4 and 3, somewhere there,
但究竟是-4和3 还是-3和4呢?
but it’s negative 4 and positive 3, or negative 3 and positive 4?
正确答案是-4和3
Ok, it’s gonna be a negative 4 and a positive 3.
因为-4乘以3等于-12
Because these multiply to give negative 12,
3减去4等于-1
but 3 take away 4 is negative 1.
接下来变换它们的符号 这就有了4和-3
Ok, so we flip the, these two numbers here, so it becomes 4 and negative 3.
再用它们除以系数6
We put them over this coefficient of 6, 6 of 6.
所以x等于什么呢?
So x can be equal to what?
x可以是2/3 4/6化简后是2/3
X could be equal to 2/3. Let’s simplify that 2/3.
x也可以是-1/2
Or x could be equal to negative 1/2.
好 这就是简单破解一元二次方程的方法
Alright. So that’s our really simple way of doing that.
你觉得如何呢?留言告诉我们吧
Tell us what you think of this.
希望你们喜欢这个视频
Hopefully you enjoyed this video.
也希望它提供给你们一些有用的信息
Hopefully it was…well, hopefully it was informative.
感谢观看 下期节目再见
Anyway, thanks for watching. See you next time. Bye.

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视频概述

想要不背公式就破解一元二次方程吗?没问题!本期节目教你如何很容易地解此类方程。敬请关注!

听录译者

sh

翻译译者

Cecilia

审核员

审核员_DY

视频来源

https://www.youtube.com/watch?v=ZGmKQa45PVw

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