Question

If alpha+beta.=5,alpha cube + beta cube=35 find the quadratic equation whose roots are alpha and beta

Answer 1

Answer:

α = 3 and β = 2

or

α = 2 and β = 3

Step-by-step explanation:

Given----> α + β = 5 and α³ + β³ = 35

To Find -----> Quadratic equation whose roots are α and β .

Solution------> We know that,

( α + β )³ = α³ + β³ + 3α β ( α + β )

Putting , α + β = 5 and α³ + β³ = 35 , we get,

=> ( 5 )³ = ( α³ + β³ ) + 3αβ ( α + β )

=> 125 = ( 35 ) + 3αβ ( 5 )

=>125 - 35 = 15 αβ

=> 90 = 15 αβ

=> αβ = 90 / 15

=> αβ = 6 ...................( 1 )

Now , α + β = 5

=> β = 5 - α

Now , putting β = 5 - α , in ( 1 ) .

=> α ( 5 - α ) = 6

=> 5α - α² = 6

=> - α² + 5α - 6 = 0

Changing sign of whole equation,

=> α² - 5α + 6 = 0

=> α² - ( 3 + 2 )α + 6 = 0

=> α² - 3α - 2α + 6 = 0

=> α ( α - 3 ) - 2 ( α - 3 ) = 0

=> ( α - 3 ) ( α - 2 ) = 0

If , α - 3 = 0

=> α = 3

Now , β = 5 - α

putting α = 3 , in it , we get,

=> β = 5 - 3

=> β = 2

If , α - 2 = 0

=> α = 2

Now, β = 5 - α

Putting α = 2 , in it , we get,

=> β = 5 - 2

=> β = 3