If roots of the equation x3 – 12 x2+39 x -28= 0 are in AP, then its common difference is-
Answers
Answer :
d = √23 ( for ascending order ) or -√23 ( for descending order )
Note :
→ The general form of a cubic equation is ; Ax³ + Bx² + Cx + D = 0 .
→ If α , ß , γ are the the roots of the cubic equation Ax³ + Bx² + Cx + D = 0 , then
• α + ß + γ = -B/A
• αß + ßγ+ γα = C/A
• αßγ = -D/A
Solution :
Here ,
The given cubic equation is ;
x³ - 12x² + 39x - 28 = 0
Now ,
Comparing the given cubic equation with the general cubic equation Ax³ + Bx² + Cx + D = 0 , we have ;
A = 1
B = -12
C = 39
D = -28
Also ,
It is given that , the roots of the given cubic equation are in AP .
Thus let α = a - d , ß = a , γ = a + d be the roots of the given cubic equation .
Now ,
The sum of roots will be ;
=> α + ß + γ = -B/A
=> (a - d) + a + (a + d) = -12/1
=> 3a = -12
=> a = -12/3
=> a = -4
Also ,
The product of roots will be ;
=> αßγ = -D/A
=> (a - d)•a•(a + d) = -(-28)/1
=> a•(a² - d²) = 28
=> -4•[(-4)² - d²] = 28
=> -(16 - d²) = 28/4
=> d² - 16 = 7
=> d² = 7 + 16
=> d² = 23
=> d = ± √23
Hence ,
• The common difference is √23 when the roots are taken in ascending order .
• The common difference is -√23 when the roots are taken in descending order .