Solving Cubic Equations with Additional Information of Roots :
Here we are going to see some example problems of solving cubic with additional information of roots.
Question 1 :
Solve the cubic equation : 2x3 − x2 −18x + 9 = 0, if sum of two of its roots vanishes
Sum of two roots vanishes means, sum of zeroes will be equal to 0.
Let the roots of the cubic equation will be α, - α and β.
From the given equation, we have
a = 2, b = -1, c = -18 and d = 9
Sum of roots = -b/a
Sum of roots (α + (- α) + β) = 1/2
β = 1/2
Product of roots = -d/a
Product of roots α (- α) β = -9/2
-α2 β = -9/2
By applying the value of β, we get
α2 (1/2) = 9/2
α2 = 9, α = ± 3
Hence the roots are 3, -3 and 1/2.
Question 2 :
Solve the equation 9x3 − 36x2 + 44x −16 = 0 if the roots form an arithmetic progression.
Let a - d, a and a + d be the roots of the equation which are in arithmetic progression.
Sum of roots = -b/a = 36/9 = 4
a - d + a + a + d = 4
3a = 4 ==> a = 4/3
Product of roots = -d/a = 16/9
(a - d) a (a + d) = 16/9
a(a2 - d2) = 16/9
By applying the value of a, we get
(4/3) ((4/3)2 - d2) = 16/9
((16/9) - d2) = (16/9) (3/4)
((16/9) - d2) = 4/3
(16/9) - (4/3) = d2
d2 = 4/9
d = ± (2/3)
If a = 4/3 and d = 2/3
a - d = 2/3
a = 4/3
a + d = 2
If a = 4/3 and d = -2/3
a - d = 2
a = 4/3
a + d = 2/3
Hence the roots are 2/3, 4/3 and 2.
After having gone through the stuff given above, we hope that the students would have understood, "Solving Cubic Equations with Additional Information of Roots".
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