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

• General_Sax
In summary: So the base must be 8.In summary, the solution of the quadratic equation x2 - 11x + 22 = 0 are x = 3 or x = 6 and the base of the numbers is 8.
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

General_Sax said:

## 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?

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

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.

## 1. What is a 'base' in a quadratic equation?

A 'base' in a quadratic equation refers to the number system being used to represent the values in the equation. This can include commonly used number systems such as binary, decimal, or hexadecimal.

## 2. How do I solve a quadratic equation with a different base?

To solve a quadratic equation with a different base, you will need to convert the values in the equation to the new base using the appropriate conversion formulas. Once all the values are in the same base, you can solve the equation using traditional methods.

## 3. Can I solve a quadratic equation using a non-integer base?

Yes, quadratic equations can be solved using non-integer bases, such as fractions or irrational numbers. However, the resulting solutions may also be non-integer values.

## 4. Is it necessary to convert the entire equation to the new base?

No, it is not necessary to convert the entire equation to the new base. You can choose to convert only the values that are in a different base, or you can convert the entire equation for consistency.

## 5. How do I know if I have solved the quadratic equation correctly in a different base?

To check if your solution is correct, you can convert the solutions back to the original base and substitute them into the equation. If the resulting values are equal, then your solution is correct.

• Precalculus Mathematics Homework Help
Replies
1
Views
1K
• Engineering and Comp Sci Homework Help
Replies
20
Views
2K
• Precalculus Mathematics Homework Help
Replies
12
Views
2K
• Engineering and Comp Sci Homework Help
Replies
10
Views
2K
• Precalculus Mathematics Homework Help
Replies
4
Views
771
• Precalculus Mathematics Homework Help
Replies
14
Views
2K
• Engineering and Comp Sci Homework Help
Replies
32
Views
3K
• Engineering and Comp Sci Homework Help
Replies
8
Views
3K
• General Math
Replies
1
Views
838
• Engineering and Comp Sci Homework Help
Replies
6
Views
2K