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
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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
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For me also same dout bro
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