If a+b=10 and ab=3, find the value of a^3+2a^2b+ab^2+100b
Answers
Given :
a + b = 10
ab = 3
To Find :
The value of a³ + 2 a² b + a b² + 100 b
Solutions :
( a - b )² = ( a + b )² - 4 a b
= ( 10 )² - 4 × 3
= 100 - 12 = 88
∴ a - b = 2 √22 .........2
∵ a + b = 10 .................1
Solving eq 1 and 2
( a + b ) + ( a - b ) = 10 + 2 √22
Or, ( a + a ) + ( b - b ) = 10 + 2 √22
Or, 2 a = 10 + 2 √22
∴ a = 5 +√22
Put the value of a into eq 1
5 +√22 + b = 10
Or, b = 10 - 5 -√22
∴ b = 5 - √22
Now,
Put the value of a and b into given equation a³ + 2 a² b + a b² + 100 b
i.e a³ + 2 a² b + a b² + 100 b = a² (a + 2 b) + b ( a b + 100)
= [ ( 5 +√22 )² { ( 5 +√22) + 2 ( 5 -√22 )} ] + [ 5 -√22 { ( 5 +√22) ( 5 -√22) + 100 } ]
= [ ( 25 + 22 + 10√22) { ( 5 +√22) + ( 10 -2√22 )} ] + [ 5 -√22 { ( 25 - 22) + 100 } ]
= [( 47 + 10√22) (15 -√22) ] + [( 5 -√22 ) { ( 3 ) + 100} ]
= [ 705 - 47√22 + 150 √22 - 220 ] + [ 15 - 3 √22+ 500 - 100√22 ]
= 485 + 103 √22 + 515 - 103 √22
= ( 485 + 515 ) + ( 103 √22 - 103 √22 )
= 1000 + 0
= 1000