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
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.
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