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.

 

[DeeNos]

I would approach this problem by first understanding the concept of bases and how they relate to numbers. A base is the number of unique digits used to represent numbers in a particular number system. For example, the decimal system has a base of 10 because it uses 10 unique digits (0-9) to represent all numbers. Binary, on the other hand, has a base of 2 because it only uses 2 unique digits (0 and 1) to represent all numbers.

In this problem, the base is not explicitly given. However, we can determine the base by looking at the solutions of the quadratic equation. We know that the solutions are x=3 and x=6. This means that the base must be a number that, when raised to the power of 3 or 6, gives us a result of 0. This is because the solutions of a quadratic equation are the values of x that make the equation equal to 0.

Based on this information, we can deduce that the base in this quadratic equation is most likely 3 or 6. However, it is not possible to determine the exact base without more information. We would need to know the values of x and r to accurately determine the base.

In terms of converting from a generic base to decimal, we can use the formula r = (a0 * b^0) + (a1 * b^1) + (a2 * b^2) + ... + (an * b^n), where r is the decimal number, b is the base, and a0, a1, a2, ... an are the digits in the number in that particular base.

In conclusion, without more information, it is not possible to accurately determine the base in this quadratic equation. However, we can use our understanding of bases and the solutions of the equation to make an educated guess.