If `a ,b , in R` such that `a+b=1a n d(1-2a b)(a ^3+b^3)=12` . The value of `(a^2+b^2)` is equal to____.

Answer

3

Explanation

Let `a^(2)+b^(2)=x`
`1-2ab=(a+b)^(2)-2ab=a^(2)+b^(2)=x`
`therefore ab=(1-x)/(2)`
`and a^(3)+b^(3)=(a+b)(a^(2)+b^(2)-ab)`
`a^(3)+b^(3)=x-ab" "(1)`
but `ab=(1-x)/(2)`
Hence `a^(3)+b^(3)=x-(1-x)/(2)=(3x-1)/(2)`
Hence the equation
`(1-2ab)(a^(3)+b^(3))=12`, becomes
`x((3x-1)/(2))=12`
`rArr3x^(2)-x-24=0`
`rArr(x-3)(3x+8)=0`
`rArrx=3 orx=-(8)/(3) ("not possible")"as " x=a^(2)+b^(2)cancellt0`
`therefore x=3 rArra^(2)+b^(2)=3`