Question

If a=2-root5/2+root5 and b=2+ root5/2-root5 find a^2-b^2

Answer 1

Answer:

a=(2-√5)/(2+√5)

={(2-√5)(2-√5)}/{(2+√5)(2-√5)}

=(2-√5)²/(2²-√5²)

=(4-4√5+5)/(4-5)

=(9-4√5)/(-1)

=-(9-4√5)

∴, a²={-(9-4√5)}²=81-72√5+80=161-72√5

b=(2+√5)/(2-√5)

=(2+√5)(2+√5)/(2-√5)(2+√5)

=(2+√5)²/(2²-√5²)

=(4+4√5+5)/(4-5)

=(9+4√5)/(-1)

=-(9+4√5)

∴, b²={-(9+4√5)}²=81+72√5+80=161+72√5

∴, a²-b²=161-72√5-161-72√5=-144√5