Find a necessary and sufficient condition on $a\in \mathbb{R}$ for a quadratic equation
$x^2+2(a+3)x-a+3=0$
to have roots such that
1)both of them are positive
2)both of them are negative

Question

anonymous · Answer

Use "\in", not "\epsilon".

The roots of the quadratic are the solutions to
$$x^2+2(a+3)\color{red}x-a+3$$
Am I right to assume there's supposed to be an $x$ there?

ujjwal · Answer

lol. yeah.. I missed x

anonymous · Answer

I left out an $=0$ at the end of that equation...

Anyway, the solutions to this equation, given by the quadratic formula, are
$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-2(a+3)\pm\sqrt{(2(a+3))^2-4(-a+3)}}{2}$$
Which values of $a$ give you both positive/both negative roots?

anonymous · Answer

$$x^2+2(a+3)x-(a-3)$$notice with roots $u,v$ we have:$$-2(a+3)=u+v\-(a-3)=uv$$if both $u,v>0$ it follows that $u+v>0$ and $uv>0$, which suggests $a+3<0\Leftrightarrow a<-3$ and $a-3<0\Leftrightarrow a<3$. note $(a<-3)\land (a<3)=a<-3$ if instead $u,v<0$ we have $u+v<0$ yet $uv>0$ suggesting $a+3>0\Leftrightarrow a>-3$ and also $a-3<0\Leftrightarrow a<3$ hence $-3

anonymous · Answer

this answer is incomplete, though; you also need that two real roots exist, which requires showing the discriminant is positive i.e.$$(2(a+3))^2-4(-(a-3))>0\4(a^2+6a+9)+4(a-3)>0\4(a^2+7a+6)>0\a^2+7a+6>0\(a+1)(a+6)>0$$

anonymous · Answer

clearly for $a>-1$ or $a<-6$ the above is satisfied hence our inequalities are actually:$$(a<-3)\land(a<-6)\implies a<-6\(-3-1)\implies -1

ujjwal · Answer

Thank you!! :)

anonymous · Answer

@SithsAndGiggles finish your approach... I want to see if it's easier :-p

anonymous · Answer

@oldrin.bataku, whether it is or not, I'm not sure, but it all comes down to solving a bunch of inequalities like with your method. Like you said, the discriminant must be positive, so you have $$4(a+3)^2+4a-12>0~~\Rightarrow~~a>-1~\text{ and }~a<-6$$ For both roots to be positive, you must have $$-2(a+3)+\sqrt{(2(a+3))^2-4(-a+3)}>0~~~~~~~~~~~~~~~~a>3~~\text{and}~~a\le-6\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow\ -2(a+3)-\sqrt{(2(a+3))^2-4(-a+3)}>0~~~~~~~~~~~~~~~a\le-6 $$ (< 0 for both to be negative roots, which gives $-1\le a<3$ for the added root and $a\ge-1$ for the subtracted root.) There seems to be a disparity with our inequality signs, but I used WA to find the solutions quickly.