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ParthKohli
 one year ago
I guess you guys will enjoy this problem (taken from Brilliant.org) 
ParthKohli
 one year ago
I guess you guys will enjoy this problem (taken from Brilliant.org) 

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ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Find the sum of all integral values of \(\alpha \in [20,20]\) such that\[2\log(x+3) = \log(\alpha x)\]has exactly one real solution.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2So first things first:\[x> 3\tag1\]\[\alpha x > 0 \tag 2 \]If we accept these two conditions, we can now recast the problem into the form.\[\log\left((x + 3)^2\right) = \log(\alpha x)\]\[\Rightarrow (x+3)^2 = \alpha x\]\[\Rightarrow x^2 + (6  \alpha)x + 9 = 0\]Pay attention to the fact that the quadratic can have one real solution, but that is not the only possibility. The quadratic may as well have two real solutions, yet one of them making the logarithm undefined. So let's first see when the quadratic itself has one real solution. I'll save us the work and tell you guys that it occurs at \(\alpha = 0, 12\) but \(\alpha = 0\) is eliminated because of the second condition of course. Now comes the hard part in my opinion.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1dw:1434998964257:dw

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2So let's try to check where \(x^2 + (6  \alpha) x + 9\) has one real root lying in the interval \((\infty, 3]\). To have two real roots, \((6  \alpha)^2 > 36\) o \(\alpha(\alpha  12) > 0\) meaning \(\alpha \in (\infty, 0) \cup (12, \infty)\). To have one root in the interval \((\infty, 3)\) ...

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2we know that \(f(3) < 0\). Thus \(9 + 9 3(6  \alpha) < 0 \Rightarrow \alpha < 0\).

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1That looks neat! for an alternative, im thinking of using calculus and mess with slopes the system has an unique solution when the slope of line is not so steep enough to cut the parabola at two different places and not too low to miss the parabola completely need to work integral values separately

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2But if \(\alpha < 0\), we also know that the other root must be in the interval \((3,0)\) since \(\alpha x>0\).

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Do we really need to solve for the above condition? We already know that the roots of the equation \(x^2 + (6  \alpha) x + 9\) have the same sign so wouldn't the second condition be covered?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2I'm not as experienced as you when it comes to calculus  but go right ahead.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2OK, \(\alpha = 20, 19, \cdots, 1, 12\) so far.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Where did Ganeshie leave?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2I'm missing steef. :(

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2I think that's it  we can't really make the \(\log(\alpha x)\) undefined for one value and defined for the other since both roots have the same sign, right? So they're in this together.

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.2Amazing, I got it right!

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1That isn't as complicated as I thought it is going to be!
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