Question

If a + b = 10 and ab = 16, find the value of a² − ab + b² and a² + ab + b²

Answer 1

In the given problem, we have to find the value of  `(a^2 - ab+ b^2),(a^2 + ab +b^2)`

Given  `a+b = 10 , ab = 16`

We shall use the identity \[\left( a + b \right)^3 = a^3 + b^3 + 3ab(a + b)\]

We can rearrange the identity as 

`a^3 + b^3 = (a+b)^3 - 3ab (a+b)`

`a^3 +b^3 = (10)^3 - 3 xx 16 (10)`

`a^3 + b^3= 1000 - 480`

`a^3 + b^3 = 520`

Now substituting values in  `a^3 + b^3 = (a+b) (a^2 + b^2 - ab)`as, `a^3 +b^3 = 520,a+b = 10`

`a^3 + b^3 = (a+b)(a^2 + b^2 - ab)`

`520 = 10 (a^2 + b^2 - ab)`

`520/10 = (a^2 +b^2 - ab)`

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

We can write  `a^2 +b^2 + ab ` as  `a^2 + b^2 +ab -2ab +2ab`

Now rearrange  `a^2+b^2+ab - 2ab +2ab` as

 `= a^2 + 2ab +b^2 -2ab +ab`

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

Thus  `a^2 +b^2 +ab =(a+b)^2 -ab`

Now substituting values  `a+b = 10,10 ab = 16` 

`a^2 +b^2 + ab = (10)^2 - 16`

`a^2 + b^2 +ab = 100 -16`

`a^2 + a^2 + ab = 84`

Hence the value of `(a^2 - ab +b^2),(a^2 + ab+b^2)`is  `52,84` respectively.