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(a x a) x 2 = (a + 5) x 81

Math brain teasers require computations to solve.

 

Puzzle ID:#4587
Fun:** (2.05)
Difficulty:** (1.93)
Category:Math
Submitted By:Cleverclogs***
Corrected By:shenqiang

 

 

 



I have an integer which, when squared and then multiplied by two is equal to that integer five higher than it multiplied by eighty one.

(a x a) x 2 = (a + 5) x 81

What is my number?

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Comments

Dazza
Jun 18, 2002

-4.5 is a solution too.
WizardMagus
Jun 19, 2002

I was about to say that too...
cleverclogs
Jun 26, 2002

Sorry, I didn't realise...
dumbell*
Sep 13, 2002

Don't be sorry. This shows that people aredoing the math, and not just clicking on answer, then rating to get points. It is still a nice teaser
fishmedAus*
Jul 02, 2003

I apparently had an error in my math and got nowhere. good one.
Sane**
Mar 13, 2005

(a * a) * 2 = (a + 5) * 81
2a x 2a = 81a + 405
2a^2 = 81a + 405
(2a^2)/2 = (81a + 405)/2
a^2 = 40.5a + 202.5
a = 1
b = 40.5
c = 202.5
x = -b{+/-}[b^2 - 4ac]^1/2 / 2a
x = -40.5{+/-}[1640.25 - 810]^1/2 / 2
x = -40.5{+/-}[28.8140509] / 2
(a - 69.3140509)(a - 11.6859491) = 0

or, a = 40.5 + 202.5/a



I don't understand, is there any real way to solve this thing? I must be missing something simple!
dishuAin*
Jun 06, 2006

Yes sane there is:
solve your equation 2a^2 = 81a+405 as follows:
2a^2-81a-405=0
=> 2a^2 -90a+9a-405=0
=> 2a(a-45)+9(a-45)=0
=> (2a+9)(a-45)=0
=> 2a + 9 =0 or a-45=0
=> a=-4.5 or a=45
pating**
Jun 19, 2007

Sane, your quadratic formula can always give you the correct answers. but you can always use factoring.
2a^2 = 81a + 405
2a^2 - 81a - 405 = 0

get the possible factors of 2 (from 2a^2), which is 2 and 1.
get the possible factors of -405, there are many like -81 and 5, -9 and 45, etc.

so you'll get :
(2a + 9)(a - 45) = 0
2a = -9; a = -4.5
or a = 45

this kind of solution gets easier through practice. and its the one being taught first before they give you the quadratic formula (at least in our math class). when all possible combinations fail, you can't go wrong with the quadratic formula.
Jimbo*au*
May 06, 2009

I liked Dishu's solution except he didn't explain how he got the (-90a + 9a) combination. It's called the sum and product method. Incidentally, I'm not sure whether the teaser has been edited but it currently asks for an integer which rules out -4.5 as a solution. As far as this being a teaser, it's question 2 part (a) from the year 9 (junior high) algebra textbook as far as I'm concerned.

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