### Brain Teasers

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

Math
Math brain teasers require computations to solve.

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?

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

What is my number?

### Hint

Square a number and then multiply it by two. Divide your answer by the number five higher than it and if the answer is less than 81 it is too low. If the answer is more than 81 it is too high.### Answer

My number is 45.(a = 45)

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## Comments

-4.5 is a solution too.

I was about to say that too...

Sorry, I didn't realise...

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

I apparently had an error in my math and got nowhere. good one.

(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!

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!

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

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

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.

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.

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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