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
Solution :
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.
Solution :
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".
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
Kindly mail your feedback to v4formath@gmail.com
We always appreciate your feedback.
©All rights reserved. onlinemath4all.com
Mar 28, 24 02:01 AM
Mar 26, 24 11:25 PM
Mar 26, 24 08:53 PM