Question

If (a-b)= -4 and an=12then find the value of
${a}^{3 } - {b}^{3}$

Answer 1

As, we know that,

(a - b)² = a² + 2ab + b²


=> a - b = 10,.

=> (a - b)² = (10)²

=> a² - 2ab + b² = 100

=> a² - 2(25) + b² = 100

=> a² - 50 + b² = 100

=> a² + b² = 100+50

=> a² + b² = 150,.....(ii),.

____________________

We know that,

(a + b)² = a² + 2ab + b²

=> (a + b)²

=> a² + 2ab + b²

=> 150 + 2(25)

=> 150 + 50

=> 200

______________

=> (a + b)² = 200

=> a + b = 10√2 ...(iii)


Adding (i) & (ii), we get,

=> a + b + a - b = 10 + 10√2

=> 2a = 10 (√2 + 1)

=> a = 5(√2 + 1),...

______________

=> a + b = 10√2

=> 5√2 + 5 + b = 10√2

=> 5 + b = 10√2 - 5√2

=> 5 + b = 5√2

=> b = 5√2 - 5

=> b = 5(√2 - 1)

__________________

We know that,

=> a³ - b³ = (a + b)(a² - ab + b²)

=> (5(√2+1) + 5(√2 -1))((5(√2 -1))² - 25 + (5(√2 - 1))²)

=> (5(√2 + 1 - √2 +1))(25(2 + 1 -2√2) - 25 + 25(2 +1 + 2√2))

=> 5 (2)( 25 ( 3 - 2√2) - 25 + 25 (3 - 2√2)

=> 5(2)( 25 (3 - 2√2 + 3 + 2√2 + 1)

=> 10(25(7))

=> 10 (175)

=> 1750,..

Answer 2

$a - b = - 4 \\ ab = 12 \\ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ {( - 4)}^{2} = {a}^{2} + {b}^{2} - 2 \times 12 \\ {a }^{2} + {b}^{2} = 16 + 24 \\ {a}^{2} + {b}^{2} = 40 \\ {a }^{3} - {b}^{3} = {a}^{2} + {b}^{2} - ab \\ {a}^{3} - {b}^{3} = 40 - 12 \\ {a}^{3} - {b}^{3} = 28 \\ \\ hope \: it \: helps \: you$