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Consider a quadratic equation x2 - 13x + 36 = 0 with coefficients in a base b. The solutions of this equation in the same base b are x = 5 and x = 6. Then b = ______.
Note: This question appeared as Numerical Answer Type in GATE.
- a)6
- b)7
- c)8
- d)9
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Consider a quadratic equation x2- 13x + 36 = 0 with coefficients in a ...
x2 – 13x + 36 = 0
==> x2 – (b + 3)x + (3*b + 6) = 0 …..
{ expanding numbers in base b )
putting the values of x we get b = 8.
==> x2 – (b + 3)x + (3*b + 6) = 0 …..
{ expanding numbers in base b )
putting the values of x we get b = 8.
Most Upvoted Answer
Consider a quadratic equation x2- 13x + 36 = 0 with coefficients in a ...
Given Quadratic Equation
The quadratic equation is x^2 - 13x + 36 = 0, with solutions x = 5 and x = 6 in base b.
Finding the Base b
Given the solutions x = 5 and x = 6, we can rewrite the quadratic equation as (x-5)(x-6) = 0.
Expanding this gives x^2 - 6x - 5x + 30 = x^2 - 13x + 30 = 0.
Comparing this with the original equation x^2 - 13x + 36 = 0, we see that the constant term is different.
The constant term in the expanded form is 30, while in the original equation it is 36.
This indicates that the base b must be such that 36 in base b is equal to 30 in base 10.
Calculating the Base b
To find the base b, we convert 36 in base b to base 10.
36 in base b = 3b + 6 = 3b + 6.
Setting this equal to 30 (the decimal equivalent), we get:
3b + 6 = 30
3b = 24
b = 8
Therefore, the base b is 8.
So, the correct answer is option C: b = 8.
The quadratic equation is x^2 - 13x + 36 = 0, with solutions x = 5 and x = 6 in base b.
Finding the Base b
Given the solutions x = 5 and x = 6, we can rewrite the quadratic equation as (x-5)(x-6) = 0.
Expanding this gives x^2 - 6x - 5x + 30 = x^2 - 13x + 30 = 0.
Comparing this with the original equation x^2 - 13x + 36 = 0, we see that the constant term is different.
The constant term in the expanded form is 30, while in the original equation it is 36.
This indicates that the base b must be such that 36 in base b is equal to 30 in base 10.
Calculating the Base b
To find the base b, we convert 36 in base b to base 10.
36 in base b = 3b + 6 = 3b + 6.
Setting this equal to 30 (the decimal equivalent), we get:
3b + 6 = 30
3b = 24
b = 8
Therefore, the base b is 8.
So, the correct answer is option C: b = 8.
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