Ta có : \(\left(a+\sqrt{a^2+2007}\right)\left(-a+\sqrt{a^2+2007}\right)=2007\)
\(\left(b+\sqrt{b^2+2007}\right)\left(-b+\sqrt{b^2+2007}\right)=2007\)
Nhân từng vế với nhau , ta có :
\(\left(a+\sqrt{a^2+2007}\right)\left(b+\sqrt{b^2+2007}\right)\left(-b+\sqrt{b^2+2007}\right)\left(-a+\sqrt{a^2+2007}\right)=2007\)
⇔ \(2007\left(-b+\sqrt{b^2+2007}\right)\left(-a+\sqrt{a^2+2007}\right)=2007^2\)
⇔ \(ab-b\sqrt{a^2+2007}-a\sqrt{b^2+2007}+\sqrt{\left(a^2+2007\right)\left(b^2+2007\right)}=2007\left(1\right)\)
Ta có : \(\left(a+\sqrt{a^2+2007}\right)\left(b+\sqrt{b^2+2007}\right)=2007\)
⇔ \(ab+a\sqrt{b^2+2007}+b\sqrt{a^2+2007}+\sqrt{\left(a^2+2007\right)\left(b^2+2007\right)}=2007\left(2\right)\)
Cộng từng vế của ( 1 ; 2 ) , ta có :
\(ab+\sqrt{\left(a^2+2007\right)\left(b^2+2007\right)}=2007\)
⇔ \(\sqrt{\left(a^2+2007\right)\left(b^2+2007\right)}=2007-ab\)
⇔ \(a^2b^2+2007a^2+2007b^2+2007^2=2007^2-2.2007ab+a^2b^2\)
⇔ \(2007a^2+2007b^2=-2.2007ab\)
⇔ \(a^2+2ab+b=0\)
⇔ \(\left(a+b\right)^2=0\)
⇔ \(S=a+b=0\)