SOLUTION: A positive number is one-half another number. The sum of the numbers is 28 less than the square of the smaller number. What are the numbers?

From the condition, it is clear that BOTH the NUMBERS are positive.

So, let x be the smaller of the two numbers; then the other number is 2x.


Next, write an equation as you read the problem

    x + 2x = x^2 - 28.


It is the same as

     3x   = x^2 - 28,

or

     x^2 - 3x - 28 = 0.


Factor left side

    (x-7)*(x+4) = 0.


The roots are 7 and -4, and because you are looking for positive solutions, 
you accept x= 7 and deny x= -4.


ANSWER.  The numbers are 7 and 14.