
Quadratic proof
I was doing some problems last night, and I came across this one and I am not sure how to get started.
If the polynomial $\displaystyle Eq. A: ax^2+bx+c$ does not have any (real*) roots, prove that $\displaystyle Eq. B: ax^2+bxc$ has two (real*) roots.
*In full disclosure, the text did not include the word 'real' before the word 'roots'. Obviously, by the Fundamental Theorem of Algebra, both will have two roots.
I was thinking about arguing that equation A must have a discriminant that is negative ($\displaystyle b^24ac < 0 $) and therefore Equation B will have a positive discriminant ($\displaystyle b^24a(c) = b^2+4ac > 0 $). A quadratic with a positive discriminant has two real roots.
Is there a better / more clever way to solve this?
Thanks

Re: Quadratic proof
Okay, but you are skipping over the crucial part of the proof: how does $\displaystyle b^2 4a(c)> 0$ follow from $\displaystyle b^2 4ac< 0$?

Re: Quadratic proof
since the quadratic ax^2+bx+c=0 has no real roots b^2<4ac .and since b^2 is positive either both a and c are positive or both are negative.In the second quadratic the c is negative therefore the product 4ac becomes now negative therefore b^2<4a(c) is not possible since b^2 is a positive number .Therefore b^24a(c)>0 and in this case the quadratic has two real roots.

Re: Quadratic proof
Thank you HallsofIvy and Minoanman!

Re: Quadratic proof
Hi Heaviside.
If it is to any help to you, I have scribbled the actual proof: http://s12.postimage.org/bpozskgal/q...rmulaproof.png
I changed letters from uppercase to lowercase. These are the same values, I just got lazy.

Re: Quadratic proof