roots of cubic equation

ROOTS OF CUBIC EQUATION

In this page roots of cubic equation we are going to see how to find relationship between roots and coefficients of cubic equation.

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

Formula:

α + β + γ = -b/a

α β + β γ + γ α = c/a

α β γ = - d/a

Example 1:

Solve the equation x³ - 12 x² + 39 x - 28 = 0 whose roots are in arithmetic progression.

Solution :

When we solve the given cubic equation we will get three roots. In the question itself we have a information that the roots are in a.p. So let us take the three roots be α - β , α , α + β

α = α - β , β = α , γ = α + β

x³ - 12 x² + 39 x - 28 = 0

ax³ + b x² + c x + d = 0

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

Sum of the roots = -b/a

α - β + α + α + β = - (-12)/1

3α = 12

α = 12/3

α = 4

The other roots can be determined by factoring the quadratic equation x² - 8x + 7

x² - 8x + 7 = 0

x² - 7x - x + 7 = 0

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

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

x -1 = 0 and x - 7 = 0

x = 1 and x = 7

Therefore the roots are 1,4,7

Example 2:

Solve the equation x³ - 19 x² + 114 x - 216 = 0 whose roots are in geometric progression.

Solution :

When we solve the given cubic equation we will get three roots. In the question itself we have a information that the roots are in g.p. So let us take the three roots be α/β , α , αβ

α = α/β , β = α , γ = α β

x³ - 19 x² + 114 x - 216 = 0

ax³ + b x² + c x + d = 0

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

Product of roots = -d/a

(α/β) x (α) x (α β) = - (-216)/1

α³ = 216

α³ = 6³

α = 6