Art of Problem Solving: 2015 AMC 10 A #23 / AMC 12 A #18

[@npip99]

3:30 That was a very interesting way to dodge long division, I never thought about it like that.

[@M_Chen333]

"9 plus 8 minus the 1/Gives us 16. And we're done." Seems Richard is just as good at poetry as he is at math.

[@tomato6844]

I know... He should go into poetry

[@yjaspal2011]

You can also use Vietna’s thereom to get the roots, r and s, are r+s=a and rs=2a. Dividing both of these expression gets us r+s/rs = 1/2. Cross-multiplying, we get 2r+2s=rs. Factoring a bit, we get Rs-2r-2s=0. Factor some more, r(s-2)-2s+4=4. In the end, we get (r-2)(s-2)=4. Solving for r and s, we get that these values of (r,s), (-1,1), (0,0),(4,4),(5,3),(6,3),(-2,1). Putting this is the equation, r+s=a, we get that a can equal 8,9, and -1. Adding them all up, we get 16 which is C.

[@devsquares]

thats what i did, but instead of dividing both equations i substituted (r+s) into a in rs=2a

[@ericzhan3454]

You could look at it by noticing that the determinant must be a square number, then checking the solutions are integers. Of course, that way you could easily miss the -1.

[@themathaces8370]

*discriminant not determinant

[@jaw5ome628]

A quicker solution would be realizing a is an integer and that you have x^2 meaning your answer would have to be a perfect square and looking at the answers you see you only have one

[@dansman1729]

non sequitur

[@זאבגלברד]

you can find from x+y=a and xy=2a that y=2x/(x-2) and this is a y=k/x that was moved 2 to the right and 2 up. and now it is simple to figure the various options for the roots,

[@zmaj12321]

Interesting shortcut to avoid long division

[@theawesomewizard4750]

What does it mean by the "zeroes of the function?"

[@rajpanchal9226]

the values at which f(x)=zero for example in function f(x)= x + 3 f(x)=0 when x = -3 so -3 is a zero of the function :)

[@sallyxu4668]

I used a pretty risky quadratic formula method & completed the square of the discriminant when I had 2 minutes left, and it actually worked :D

[@themathaces8370]

Just get that a^2-8a=a(a-8) is a perfect square by discriminant. 9,8 obviously work. Then we find that -1 works as well, so 16.

[@jdoan2017]

Simon's Favorite Factoring Trick!!!: www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=623889

[@helo3827]

I just use vietas and bashed.