Expansion and factorization to determine roots of equations

Question

Let $(a,c)$ be the roots of the equation $x ^ 2 + ax - b = 0$.

Let $(b,d)$ be the roots of the equation $x ^ 2 + cx + d = 0$.

Find all the possible real values for $a, b, c, d$.

NOTE: I have made very little progress towards the answer and any partial answers/hints/explanations/ answers would be extremely helpful. Thanks in advance. :)

score 1 · Answer 1 · answered Jun 29 '15 at 09:33

1

HINT:

Immediately we have $a+c=-a\iff c=-2a$ and $bd=d\iff d(b-1)=0$

and $ca=-b\implies(-2a)a=-b\implies b=2a^2$

and $b+d=-c\implies2a^2+d=-2a\iff d=-2a-2a^2$

If $d=0,-2a-2a^2=0$

If $b-1=0\iff b=1,2a^2=1$

answered Jun 29 '15 at 09:33

lab bhattacharjee

274,582

@Jupiter He's using here the fact that, for any 2nd-degree polynomial expression $\alpha x^2+\beta x+\gamma$, the sum of the roots is equal to $-\beta / \alpha$, and the product is equal to $\gamma / \alpha$ – BusyAnt Jun 29 '15 at 09:41
@BusyAnt, formally known as https://en.wikipedia.org/wiki/Vieta's_formulas – lab bhattacharjee Jun 29 '15 at 09:43
You're totally right, but I thought this needed to be made clear and visible right next to the answer – BusyAnt Jun 29 '15 at 09:51
Thanks!! This was helpful! – Carli Jun 30 '15 at 10:48

Expansion and factorization to determine roots of equations

1 Answers1