Quadratic Equations And Cubic Equations Worksheet With Answers Page 4
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Cubic Equations
2.1
Introduction
The general form of a cubic equation is
3
2
ax
+ bx
+ cx + d = 0
(2.1)
where a,b,c and d are constants, a = 0.
Equation (2.1) is also expressible as
b
c
d
3
2
x
+
x
+
x +
= 0
(2.2)
a
a
a
If α,β and γ are roots of the cubic equations (2.1), (2.2), then
b
c
d
3
2
x
+
x
+
x +
(x
α)(x
β)(x
γ)
a
a
a
3
2
= x
(α+β+γ)x
+(αβ+βγ+γα)x αβγ
Thus comparing coefficients,
b
α + β + γ =
a
c
αβ + βγ + γα =
a
d
αβγ =
.
a
Thus the equation whose roots are α,β,γ is
3
2
x
(α + β + γ)x
+ (αβ + βγ + γα)x
αβγ = 0
(2.3)
2.2
Useful identities and examples
2
2
2
2
α
+ β
+ γ
= (α + β + γ)
2(αβ + βγ + γα)
3
3
3
3
α
+ β
+ γ
= (α + β + γ)
3(α + β + γ)(αβ + βγ + γα) + 3αβγ.
3
Example 2.1. If α,β and γ are the roots x
7x + 1 = 0, find the equation
2
2
2
whose roots are α
,β
,γ
.
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