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

[General_Sax]

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

Please help me.

 

[Mark44]

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 = 1_b
-9₁₀ = -11_b
18₁₀ = 22_b

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.

 

[General_Sax]

It's base 8 right?

 

[Mark44]

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