Question

hello every one .
I am posting a tutorial on : The topic of quadratic equation : roots of quadratic equation

Answer 1

So , Today i am going to post a tutorial on A theorem on quadratic equations :

Have you ever thought of that is their any proof of : Can a quadratic equation only have 2 roots , cant it more ? There is a simple reason that is the degree of polynomial is 2 . But I
am going to post the exact proof for this : (* 2 distinct real roots )


Quadratic equation general formula ----------> \(\large{ax^2+bx+c=0}\)

So let us have : three roots : \(\alpha\) , \(\beta\) , \(\gamma\) : alpha , beta and gamma are distinct roots and none of them equals zero .

If alpha , beta and gamma are the roots then it is necessary that:

\(\large{a(\alpha^2)+b(\alpha)+c=0}\) ------------(1)
\(\large{a(\beta^2)+b(\beta)+c=0}\) ------------(2)
\(\large{a(\gamma^2)+b(\gamma)+c=0}\) ------------(3)

So first of all we will subtract eq. 2 from eq. 1 that is :

\[\large{a(\alpha^2)+b(\alpha)+c-a(\beta^2)-b(\beta)-c=0}\]
\[\large{a(\alpha^2)-a(\beta^2)+b(\alpha)-b(\beta)+c-c=0}\]
\[\large{a(\alpha^2-\beta^2)+b(\alpha-\beta)=0}\]
\[\large{a(\alpha+\beta)(\alpha-\beta)+b(\alpha-\beta)=0}\]
Applying inverse of distributive property we get :
\[\large{(\alpha-\beta)[a(\alpha+\beta)+b]=0}\]
\(\large{a(\alpha+\beta)+b=0}\)---------------(4)
SINCE : we have taken alpha and beta as two distinct roots , hence their diff. can not be 0 .

Now similarly subtracting eq. (3) from eq. (2)

\[\large{a(\gamma^2)+b(\gamma)+c-a(\beta^2)-b(\beta)-c=0}\]
\[\large{a(\gamma^2)-a(\beta^2)+b(\gamma)-b(\beta)+c-c=0}\]
\[\large{a(\gamma^2-\beta^2)+b(\gamma-\beta)=0}\]
\[\large{a(\gamma+\beta)(\gamma-\beta)+b(\gamma-\beta)=0}\]
\[\large{(\gamma-\beta)[a(\gamma+\beta)+b] = 0}\]
\(\large{a(\gamma+\beta)+b=0}\) ------------(5)
SINCE : We have taken gamma and beta as two distinct roots , hence their diff. can not be 0

Now Subtracting eq.(5) from eq.(4) :

\[\large{a(\alpha+\beta)+b-a(\gamma+\beta)-b=0}\]
\[\large{a(\alpha+\beta)-a(\gamma+\beta)+b-b=0}\]
\[\large{a(\alpha+\beta-\gamma-\beta)=0}\]
\[\large{a(\alpha-\gamma)=0}\]

But this is not possible : Because alpha and gamma are distinct roots and alpha is not equal to zero . So , their product can not be zero . Thus the assumption that a quadratic equation has three distinct real roots is wrong . Hence a quadratic equation can not have more than 2 roots.

Answer 2

awesome @mathslover :)

Answer 3

and \(a\) cannot be zero or else the equation would not be a quadratic

Answer 4

yes thanks for that point also @UnkleRhaukus

Answer 5

Great work mathslover this tutorial is goood/.

Answer 6

thanks @annas and @UnkleRhaukus I will post some more tutorials soon whenever i will get time

Answer 7

welcome maths :) wish you best of luck for future :)

Answer 8

How do you guys post such long tutorials so quickly?

Answer 9

i just make it ready and then post this

Answer 10

I see ... that's actually quite a good idea.

Answer 11

Good to hear that u r also making some tutorials best of luck :
thanks also

Answer 12

So friends here is my whole tutorial in a drawing form : i did it on my own

Answer 13

@dpaInc , @eashmore , @jhonyy9 have a look to this tutorial :)

Answer 14

@vishweshshrimali5 and @Zarkon sir please give some suggestions

Answer 15

That is impressive.

Answer 16

Thanks mam .. I will post some more soon .. i just need support from u all :)

Answer 17

KV rocks!

Answer 18

Kendriya Vidyalaya

Answer 19

Thanks a lot mam ! .. Thanks to my teachers and to all of u also .... :)

Answer 20

Your equation \(a(\alpha-\gamma)=0\) simply implies \(\alpha=\gamma\), thus confirming there are only two roots: \(\alpha\) and \(\beta\). (This is just how I looked at it. After reading your entire tutorial, your method makes sense.)

Answer 21

Yes @Limitless thanks a lot for adding this to my tutorial ... this thing looks good :
\[\huge{\alpha-\gamma=0}\]
\[\huge{\alpha=\gamma}\] Only if alpha is not equal to 0 that is given to us.. thanks again

Answer 22

It doesn't matter what \(\alpha\) is; it's true that \(\alpha=\gamma\) regardless.

Answer 23

K ! thanks

Answer 24

Really nice work.............Well done!

Answer 25

thanks a lot @nitz

Answer 26

i must say your tutorials are really nice and useful!

Answer 27

http://openstudy.com/study#/updates/4ff50f22e4b01c7be8c837ca have a look to my new tutorial

Answer 28

Nice

Answer 29

Ehh :p
http://openstudy.com/users/.sam.#/updates/4fb7bd6fe4b05565342d33a0

Answer 30

:p medals level :)