Question

a=2-√5÷2+√5 and b=2+√5÷2-√5,find a²-b².

Answer 1

a=2-√5/2+√5
rationalize the denominator
a=2-√5/2+√5×2-√5/2-√5
a=(2-√5)^2/(2)^2-(√5)^2
a=(2)^2+(√5)^2-2×2×√5/4-5
a=4+5-4√5/-1
a=-9-4√5/-1
=-(-9-4√5)/1
=9+4√5
b=2+√5/2-√5
rationalize the denominator
b=2+√5/2-√5×2+√5/2+√5
=(2+√5)^2/(2)^2-(√5)^2
=(2)^2+(√5)^2+2×2×√5/4-5
=4+5+4√5/-1
=9+4√5/-1
=-(9+4√5)/1
=-9-4√5
now ,
a^2-b^2=(9+4√5)^2 -(-9-4√5)^2
={(9)^2+(4√5)^2+2×9×4√5 } -{(-9)^2+(4√5)^2-2×-9×4√5}
=(81+80-72√5)-(81+80+72√5)
=(161-72√5)-(161+72√5)
=161-72√5-161-72√5
=161-161-72√5-72√5
=-144√5