Problem in finding quad. eqn. from the roots

[Sumedh]

In general, if α(alpha) and ß (beta) are roots of eqn. ax^2 +bx +c=0
then for finding the equation whose roots are α+2 and ß+2 can be done by
addition of roots (α+2+ß+2=-b/a) and product of roots (α+2)(ß+2)=c/a
By solving this we get ax^2 -(4a-b)x + (4a-2b+c)=0

The problem is this that,by replacing x in place of (x-2)in the given equation
ax^2 +bx +c=0 we get the same answer ax^2 -(4a-b)x + (4a-2b+c)=0
but this method ( replacing x by (x-2) ..) is not mentioned anywhere
i am not able to understand this method
please help .

 

[HallsofIvy]

I'm not sure what you are asking. Certainly, it is true that if $\alpha$ and $\beta$ are roots of a quadratic equation, then the equation is of the form $a(x- \alpha)(x- \beta)= 0$ for some number a.

Similarly, if the roots of a quadratic equation are $\alpha+ 2$ and $\beta+ 2$, then the equation is of the form $a(x- (\alpha+ 2))(x- (\beta+ 2)= 0$ which is the same as $a((x- 2)- \alpha)(x- 2)- \beta)= 0$, the original equation with "x" replaced by "x- 2".

 

[Sumedh]

thanks i got it