Quadratic Equations: Area of Quadrilateral Calculation

[erisedk]

Homework Statement

If a, b, c, d are the positive roots of x^4 - 12x^3 - px^2 + qx + 81 = 0, then the area of the quadrilateral formed by x=a, x= -b , y = c and y = -d is:
Ans: 36

Homework Equations

Vieta, I guess.

The Attempt at a Solution

I know its going to form a rectangle with sides (a+b) and (c+d). The area would be (a+b)(c+d). The only information I have is a+b+c+d=12 and abcd=81. Somehow, I need to convert that into ac+ad+bc+bd. I have no idea how. Introducing p for ab+bc+cd+da seems useless because even that doesn't help much.

 

[haruspex]

Check the sanity of the question by trying a couple of sets of values. The obvious 3,3,3,3 works, so try something else. If the claim appears true this might start to shed light on why.

 

[Linghunt]

Slap the guy that wrote problem. Not even sure what you are trying to solve or do. Quadrilateral is in 1 plane ( so no 3-D) and 4 sides and length are open.

Factoring out the equation would be 1st step, Perhaps the roots from that will give you numbers to get the 36. The callout for x=a, etc makes no sense. perhaps this is in an example problem in test for jargon. What class is this for?

Haruspex, seems to read it ok, perhaps it's just me. Slap the writer anyway. ; /

 

[erisedk]

The four sides are not open. The four lines will intersect at four points and form a quadrilateral, parts of which will lie in all four quadrants. So, that is perfectly valid. I can't really factorise a biquadratic equation, so I can't get the roots directly.
Oh, and this is from a test in a preparatory course for JEE, which is an entrance exam for certain engineering institutes that about 1.2 million 12th grade students take each year in India, and only 2-3% get into the above mentioned colleges. So, it's supposed to be hard, though I would loooovve to slap the writer anyway ;)

Check the sanity of the question by trying a couple of sets of values. The obvious 3,3,3,3 works, so try something else. If the claim appears true this might start to shed light on why.

When I think about it, 81 only has 3 as a prime factor and 9 as the only other factor. 9+1+1+1 gives 12 but the product isn't 81. So, 3,3,3,3 seem to be the only option.

 

[erisedk]

OMG I got it. It isn't even a quadratic problem. It's an AM-GM inequality problem!
(a+b+c+d)/4 >= (abcd)^1/4

In this case, equality holds.
The only case in which equality holds is when all the four numbers are equal, which is 3, 3, 3, 3 for the fact that a+b+c+d = 12.
Therefore, area is 36!
Thank you guys :D

 

[Linghunt]

Kool , Glad I helped, a little. I was a pain in the butt at times in school, I loved a problem that did not provide enough info, so I got to assume values, I liked assigning like ZERO or infinite to make answer easy. Got some good arguments but I won most of them, Got better questions on next test.

If you don't have Arthur of test in line of sight, not a good method. Still wasn't a real good idea, but was fun.

 

[Linghunt]

Oh, and this is from a test in a preparatory course for JEE, which is an entrance exam for certain engineering institutes that about 1.2 million 12th grade students take each year in India, and only 2-3% get into the above mentioned colleges.

I interviewed a guy with EE degree from India years ago, he was looking to get E Tech job, Pulling wire and reading my schematics. I was going to give him thumbs up to get foot in door, then I learned he could do a voltage divider... I figured he was liar or really stupid.

I learned those in junior high before high school, and I'm one of the duller knifes in the box.

 

[erisedk]

If you don't have Arthur of test in line of sight, not a good method. Still wasn't a real good idea, but was fun.

What does Arthur of test in line of sight mean? I assume it's a typo but what is it?

I interviewed a guy with EE degree from India years ago, he was looking to get E Tech job, Pulling wire and reading my schematics. I was going to give him thumbs up to get foot in door, then I learned he could do a voltage divider... I figured he was liar or really stupid.

I learned those in junior high before high school, and I'm one of the duller knifes in the box.

To be very honest, most colleges (and schools ) in India are pretty useless at teaching you anything. The IITs, which are the colleges that require you to give JEE(Advanced), are one of the very few that are at least decent~good. That is why the competition to get into IIT is so intense. That is why so many kids from India want to study abroad.

 

[Linghunt]

No typo. line of sight means he is in front of you to voice your comments and smell your breath. Person to person.

Line of sight reference to hunting and firearms, We still have ours here.

JEE is Electrical Engineering right? I can give you intell so you can do circuits by inspection. That means you can write the answer in one step.

Can you do Ohms law, KCL, KVL , superposition? and basic alegbra. All in one step. I learned this as a Junior in BSEE classes, I said why did I not learn this as a freshman... Politics and Book sales.

This is not me, but a mentor of mine. You apply yourself, and read material you do well. Let me know.

 

[erisedk]

Oh no! JEE is Joint Entrance Examination for admission into the Indian Institutes of Technology. I'm in 11th grade and it is for admission into engineering and pure math and science courses :)

 

[erisedk]

Thanks for the offer anyway, though I plan to major in Computer Science and Eng or Mathematics :)

 

[Linghunt]

Understand, When you are figuring out a voltage and what is going on and scratching your head, perhaps you will remember.

You need to be a Math Wiz to be a good software guy or Engineer. Can you count in base 16 as easy as you can in base 10. Subtract number on the fly and understand, not just pocket Calculator.