## ROOTS OF CUBIC EQUATION

Let ax3 + bx2 + cx + d = 0 be any cubic equation and α, β  and γ be the roots.

α + β + γ = -b/a

αβ + βγ + γ α = c/a

αβγ = -d/a

Example 1 :

Solve the following cubic equation whose roots are in arithmetic progression.

x3 - 12x2 + 39 x - 28 = 0

Solution :

When we solve cubic equation we will get three roots.

Since the roots are in arithmetic progression, the roots can be taken as given below.

p - q, p, p + q

Compare :

x- 12x+ 39x - 28 = 0 and ax+ bx+ cx + d = 0

a = 1, b = -12, c = 39, d = -28

Sum of the roots = -b/a

p - q + p + p + q = -(-12)/1

3p = 12

p = 4

4 is one of the roots. The other roots can be determined by solving the quadratic equation

x- 8x + 7 = 0

x- 8x + 7 = 0

(x - 1)(x - 7) = 0

x - 1 = 0 or x - 7 = 0

x = 1 or x = 7

Therefore the roots are 1, 4 and 7.

Example 2 :

Solve the following cubic equation whose roots are in geometric progression.

x- 19x2 + 114x - 216 = 0

Solution :

When we solve cubic equation we will get three roots.

Since the roots are in geometric progression, the roots can be taken as given below.

p/q, p, pq

Compare :

x- 19x+ 114x - 216 = 0 and ax+ bx+ cx + d = 0

a = 1, b = -19, c = 114, d = -216

Product of roots = -d/a

p/q  p  pq = -(-216)/1

p3 = 216

p3 = 63

p = 6

6 is one of the roots. The other roots can be determined by solving the quadratic equation

x- 13x + 36 = 0

(x - 4)(x - 9) = 0

x - 4 = 0 or x - 9 = 0

x = 4 or x = 9

Therefore the roots are 4, 6 and 9.

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