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 $Eq. A: ax^2+bx+c$ does not have any (real*) roots, prove that $Eq. B: ax^2+bx-c$ 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 ( $b^2-4ac < 0$) and therefore Equation B will have a positive discriminant ( $b^2-4a(-c) = b^2+4ac > 0$). A quadratic with a positive discriminant has two real roots.

Is there a better / more clever way to solve this?

Thanks

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

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^2-4a(-c)>0 and in this case the quadratic has two real roots.

Thank you HallsofIvy and Minoanman!