Question

if a and b are unequal and x^2+ax+b and x^2+bx+a have a common factor, then show that a+b+1=0​

Answer 1

Answer:

let common factor is y

 so y = [ -a  +_sqrt(a^2- 4b)/2 = [ -b+- sqrt( b^2-4a]/2

or -a + sqrt( a^2- 4b) = -b + sqrt(b^2- 4a}

 ( a -b) = sqrt(a^2 -4b) - sqrt(b^2-4a)

or a^2- 2ab + b^2 = a^2 - 4b + b^2- 4a - 2sqrt(a^2-4b) (b^2-4a)

  or -2ab + 4b+ 4a = - 2 sqrt(a^2-4b) (b^-4a)

   (-ab+2b+2a)^2 = (a^2-4b) (b^2-4a)

  a^2b^2+4b^2+4a^2-4ab^2-4a^2b+8ab= a^2b^2-4a^3 - 4b^3+ 16ab

   a^3+ b^3-2ab+a^2+b^2-b^2a-a^2b =0

  a^3 + b^3 - ab( a+b) +( a^2+ b^2 -2ab) =0

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

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

       (a+b) ( a-b)^2 =-  (a-b)^2  

       or  a+b= - 1               [ as a-b is not equal to zero ]

     or   a+b+ 1 =0 proved

#Capricorn Answers

Answer 2

For me also same dout bro