solve the following equation given that each root is in AP
Answers
Answer :
-2 , 1 , 4
Solution :
★ The general form of a cubic equation is ; ax³ + bx² + cx + d = 0 .
★ Let α , ß and γ be the roots of the cubic equation ax³ + bx² + cx + d = 0 , then
• Sum of roots , α + ß + γ = -b/a
• Sum of roots taking two at a time ,
αß + ßγ + αγ = c/a
• Product of roots , αßγ = -d/a
★ A.P. (Arithmetic Progression) : A sequence in which the difference between the consecutive terms are equal is said to be in A.P.
★ If a1 , a2 , a3 , . . . , an are in AP , then
a2 - a1 = a3 - a2 = a4 - a3 = . . .
★ Three numbers in AP can be given as ;
a - d , a , a + d .
★ Four numbers in AP can be given as ;
a - 3d , a - d , a + d , a + 3d .
Solution :
Here ,
The given cubic equation is ;
x³ - 3x² - 6x + 8 = 0 .
Comparing the given cubic equation with the general cubic equation
ax³ + bx² + cx + d = 0 , we have ;
a = 1
b = -3
c = -6
d = 8
Also ,
It is given that , the roots of the given cubic equation are in AP .
Thus ,
Let the roots of the given cubic equation be ; a - d , a , a + d .
Now ,
=> Sum of roots = -b/a
=> (a - d) + a + (a + d) = -(-3)/1
=> 3a = 3
=> a = 3/3
=> a = 1
Now ,
=> Product of roots = -d/a
=> (a - d)×a×(a + d) = -8/1
=> a(a - d)(a + d) = -8
=> a(a² - d²) = -8
=> 1(1² - d²) = -8
=> 1 - d² = -8
=> 1 + 8 = d²
=> 9 = d²
=> d² = 9
=> d = √9
=> d = ±3
★ If a=1 and d=3 , then the roots of the given cubic equation will be ;
• a - d = 1 - 3 = -2
• a = 1
• a + d = 1 + 3 = 4
★ If a=1 and d=-3 , then the roots of the given cubic equation will be ;
• a - d = 1 - (-3) = 1 + 3 = 4
• a = 1
• a + d = 1 + (-3) = 1 - 3 = -2