# SOLVING CUBIC EQUATIONS WITH ADDITIONAL INFORMATION OF ROOTS

## About "Solving Cubic Equations with Additional Information of Roots"

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.

## Solving Cubic Equations with Additional Information of Roots - Practice questions

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

-αβ  =  -9/2

By applying the value of β, we get

α(1/2)  =  9/2

α =  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

d=  4/9

d  =  ± (2/3)

 If a = 4/3 and d = 2/3a - d  =  2/3a  =  4/3a + d  =  2 If a = 4/3 and d = -2/3a - d  =  2a  =  4/3a + 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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