ROOTS OF CUBIC EQUATION

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 :

x³ - 12x² + 39x - 28 = 0 and ax³ + bx² + 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

x² - 8x + 7 = 0

x² - 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. 

x³ - 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 :

x³ - 19x² + 114x - 216 = 0 and ax³ + bx² + 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

x² - 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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