Question

if alpha+bita=5 and alpha^3+bita^3=35
find the quadratic equation whose roots are alpha and bita

Answer 1

alpha+ beta=5
alpha^3+beta^=35
By solving alpha=2
beta=3
Therefore
P(x)=x^2-(sum of roots)x+product of roots
       =x^2-5+6
hope this is correct

Answer 2

Given,
α+β = 5 and,
α³+β³ = 35

(a+b)³ = a³+b³+3ab(a+b)
Here, a=α and b=β
(α+β)³ = α³+β³+3αβ(α+β)
Entering the given values, we get
(5)³ = 35 + 3αβ(5)
125 = 35 + 3αβ(5)
90 = 3αβ(5)
αβ = 6

The quadratic equation is given by
⇒ x²-(α+β)x+αβ
∴ x²-5x+6