Equation x2−ax+b and x2+bx−a have a common root. Let alpha be the common root of both quadratic equations ⇒α2−aα+b=0 α2+bα−a=0 Using cross-multiplication
α2
−a
b
b
−a
=
α
b
1
−a
1
=
1
1
−a
1
b
α2
a2−b2
=
α
b+a
=
1
b+a
We get
α
b+a
=
1
b+a
⇒α=1 Now, using first two fraction
1
a2−b2
=
1
a+b
⇒a+b=a2−b2 ⇒(a+b)=(a+b)(a−b) ⇒(a+b)(a−b−1)=0 a+b=0 or a−b−1=0 ⇒a−b=1