Question

If a+b=5 and a^2+b^2=11 then prove that a^3+b^3=20?

Answer 1

a^2 + b^2 = (a+b)^2 - 2ab

(a+b)^2 - 2ab = 11

5^2 - 2ab = 11

25 -2ab = 11

-2ab = -14

ab = 7

now

a^3+b^3 = (a + b)(a^2 + b^2 – ab)

= (5)(11-7)

= 5 * 4

= 20

Answer 2

Answer:

The proof is given in the solution below :

Step-by-step explanation:

Given,

a + b = 5

a² + b² = 11

Now we know,

( a + b )² = a² + b² + 2ab

Substituting the value of ( a + b ) and a² + b²

( 5 )² = 11 + 2ab

25 = 11 + 2ab

25 - 11 = 2ab

2ab = 14

Dividing both sides by 2,

2ab / 2 = 14 / 2

ab = 7

Now,

a³ + b³ = ( a + b ) ( a² -ab + b² )

a³ + b³ = ( a + b ) ( a² + b² - ab )

Substituting the values,

a³ + b³ = ( 5 ) ( 11 - 7 )

a³ + b³ = ( 5 ) ( 4 )

a³ + b³ = 20

Hence,

Proved that if a+b=5 and a^2+b^2=11 then a^3+b^3=20