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 =

                      α = 6

The other roots can be determined by factoring the quadratic equation x² - 13x + 36

x² - 13x + 36 = 0

x² - 9x  - 4x + 36 = 0

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

  (x - 4) (x - 9) = 0

  x - 4 = 0  and x - 9 = 0

   x = 4 and x = 9

Therefore the roots are 4,6,9

These are the examples of roots of cubic equation.

Related Topics




Roots of Cubic Equation to Algebra