Solve by using the perfect squares method.
(2x + 4)2 = 64

Let x be the number of women

y be the number of men

a total of 416 adult beverages were sold

So x + y = 416   equation 1

all adult are $ 1.25 for women and $ 2.50 for men and sold a total of $ 997.50.

So 1.25x + 2.50y = 997.50   equation 2

Now solve both the equations for number of women that is x

using first equation : x + y = 416

Subtract x on both sides . y = 416 -x

Now replace y in second equation

1.25x + 2.50y = 997.50

1.25x + 2.50(416 -x) = 997.50

1.25x + 1040 - 2.50x = 997.50

-1.25x + 1040 = 997.50 (subtract 1040 on both sides)

-1.25x = -42.5 (divide by -1.25 on both sides)

x= 34

There are 34 adult beverages were sold to​ women.