Question

if alpha plus beta is equal to 5 and Alpha Cube + beta cube is equal to 53 find the quadratic equation​

Answer 1

Step-by-step explanation:

Given---> α + β = 5 and α³ + β³ = 53

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

Solution---> We have an identity

a³ + b³ = ( a + b ) ( a² + b² - ab )

Applying it ,

α³ + β³ = ( α + β ) ( α² + β² - αβ )

Putting α + β = 5 in it we get , and adding and subtracting 2αβ to make perfect square.

=> 53 =( 5 ) {( α² + β² + 2αβ ) -2αβ -αβ }

We have an identity

(a + b )² = a² + b² + 2ab , applying it here

=> 53 / 5 = ( α + β )² - 3αβ

=> 53 / 5 = ( 5 )² - 3 αβ

=> 53 / 5 = 25 - 3 αβ

=> 3αβ = 25 - ( 53 / 5 )

=> 3αβ = (125 - 53) / 5

=> 3αβ = 72 / 5

=> αβ = 72 / (3 × 5)

=> αβ = 24 / 5

Required quadratic equation

x² - (sum of roots)x+ product of roots=0

=> x² - (α + β )x + ( αβ ) = 0

=> x² - ( 5 ) x + ( 24 / 5 ) = 0

=> 5x² - 25x + 24 = 0