# Solving for a 'base' (eg binary) in a quadratic equation

## Homework Statement

The solution of the quadratic equation x2 - 11x + 22 = 0 are x = 3 or x = 6. What is the base of the numbers?

## Homework Equations

Knowledge of how to convert from a generic base to decimal?

## The Attempt at a Solution

I tried to just place r in where I would have a value of s*r1

(x-3)(x-6) = x2 - 11x + 22

x2 - 6x - 3x + [1*r + 8] = x2 - [x( r + 1)] + 2r + 1

r + 8 - 9x = x + 2r - xr + 1

7 - 10x = r(1-x)

r = (7-10x)/(1-x)

when I try to put either of the values of x in I get r as either 10.6 or 11.5

Mark44
Mentor

## Homework Statement

The solution of the quadratic equation x2 - 11x + 22 = 0 are x = 3 or x = 6. What is the base of the numbers?

## Homework Equations

Knowledge of how to convert from a generic base to decimal?

## The Attempt at a Solution

I tried to just place r in where I would have a value of s*r1

(x-3)(x-6) = x2 - 11x + 22

x2 - 6x - 3x + [1*r + 8] = x2 - [x( r + 1)] + 2r + 1

r + 8 - 9x = x + 2r - xr + 1

7 - 10x = r(1-x)

r = (7-10x)/(1-x)

when I try to put either of the values of x in I get r as either 10.6 or 11.5
The base should be an integer.

Since 3 and 6 are roots of the equation, it's safe to assume that the base is at least 6.
Also, since 3 and 6 are roots, x - 3 and x - 6 are factors of the quadratic.

On the one hand you have (x - 3)(x - 6) = x2 - 9x + 18 (in base-10).
On the other hand, you have x2 - 11x + 22 (in unknown base).

Comparing the coefficients of the first expression with the second, you must have
110 = 1b
-910 = -11b
1810 = 22b

What does b need to be so that all three equations are true statements?
Note that d1d2 in base b = d1 * b + d2 in base 10.

It's base 8 right?

Mark44
Mentor
Right. Notice that 118 means 1*8 + 1*1 = 910, and 228 means 2*8 + 2*1 = 1810.