Final answer:
To solve the problem, two equations are formed based on the given conditions: one representing the relationship between the two numbers and the other their product. The equations are solved by substitution and factoring, leading to a pair of positive numbers that satisfy both equations.
Step-by-step explanation:
We have a mathematical problem that involves forming equations based on the given conditions and solving them to find two positive numbers. According to the problem statement, let's denote the smaller positive number as x, and the larger one as y. We are given that one positive number is 8 more than twice the other, which can be expressed as y = 2x + 8. Also, the product of these two numbers is 24, leading to our second equation, xy = 24.
Substituting the expression for y from the first equation into the second, we get x(2x + 8) = 24. This simplifies to 2x² + 8x - 24 = 0. We can then factor this quadratic equation to find the values of x. After finding values for x, we can substitute back into the first equation to find the corresponding values of y.
Solving the factored quadratic equation, we find two possible solutions for x, but since we are looking for positive numbers, we take the positive solution for x. With this value, we easily find the positive value for y using our first equation. Therefore, we end up with two positive numbers that satisfy both initial conditions.