# finding base of quadratic system

#### bhuvanesh

Joined Aug 10, 2013
268
The roots of the quadratic equation x^2-12x+37=0 are 5 and 8. Find the base system in which this equation is written

If the roots are 5 and 8, then the equation can be written as
(((x^2 -(5+8)x + (5*8) = 0}}}
so in what base is 5+8 = 12 and 5*8 = 37?
The base must be greater that 8 since one of the solutions is 8.
The base is greater than 10 since 5*8 base 10 is 40 and our product is 37
Try 11. Hey, it works!
5+8 base 11 = 12 and (5*8) base 11 = 37.

How they say the base should be greater than 8 and also than 10..thank you in advance

#### shteii01

Joined Feb 19, 2010
4,644
I got different roots.

#### djsfantasi

Joined Apr 11, 2010
9,188
Since one of the roots is 8, the base must be greater than 8. What if the base were less than 8? Then there is no digit 8! A value of decimal 8 in base 8 would be 10!

Similar logic can be applied for saying that the base must be greater than 10. 5*8=40 in base 10. But the result is 37. If the base were 10 or less, the result is 40 or more. What is 40 base 10 in base 9? (44) See the direction the digits are going?

It can be proved mathematically, but observation satisfied me.

#### djsfantasi

Joined Apr 11, 2010
9,188
Shteii01 and PsySc0rpi0n:

Are you finding the decimal roots? Note the problem states that the numbers are represented in a base other than base 10? Using base 11 numbers, 5 and 8 are the roots of the equation.

#### PsySc0rpi0n

Joined Mar 4, 2014
1,777
The roots of the quadratic equation x^2-12x+37=0 are 5 and 8.
Shteii01 and PsySc0rpi0n:

Are you finding the decimal roots? Note the problem states that the numbers are represented in a base other than base 10? Using base 11 numbers, 5 and 8 are the roots of the equation.
I see... But that's not what the OP seems to say in the statment above... Even though, is it or is it not important that real roots are, in fact, not real but imaginary roots?

#### shteii01

Joined Feb 19, 2010
4,644
Shteii01 and PsySc0rpi0n:

Are you finding the decimal roots? Note the problem states that the numbers are represented in a base other than base 10? Using base 11 numbers, 5 and 8 are the roots of the equation.
Ah. I see what you are saying. The roots 5 and 8 are given to the OP.