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If roots of the equation X³-12x²+39x-28=0 are in AP then the common differences is A ±4 B ±3 C±2 D ±1 What is the formulas for difference of root

If roots of the equation X³-12x²+39x-28=0 are in AP then the common differences is
A ±4
B ±3
C±2
D ±1
What is the formulas for difference of root

Grade:11

1 Answers

Arun
25750 Points
5 years ago
The roots are 1,4, and 7, and arithmetic progression with a difference of 3. 

There are 3 roots that add up to -b. That means if the common difference is d and the 3 consecutive terms of the arithmetic progression are x - d, x , and x + d, then x - d + x + x + d = 12. This simplifies to 3x = 12 and x = 4. the middle of the 3 numbers is 4. 

Since all 3 numbers must multiply together to equal the opposite of the constant, 28, then (4)times the other 2 roots = 28. That means the other 2 roots have to multiply to 7. The only integer values are 1 times 7 and these form an arithmetic sequence along with 4. So the roots are 1, 4, and 7.
 
hence common difference is ± 3

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