10. The larger of two positive integers is three more than the smaller. If twice the
square of the smaller integer is increased by three times the larger integer, the
result is 74. Find the integers.

The integers are 8 and 5

Step-by-step explanation:

An integer is a whole number.

Let x represent the larger positive integer.

Let y represent the smaller positive integer.

The larger of two positive integers is three more than the smaller. It means that

x = y + 3

If twice the square of the smaller integer is increased by three times the larger integer, the result is 74. It means that

2y²+ 3x = 74- - - - - - - - - - -1

Substituting x = y + 3 into equation 1, it becomes

2y² + 3(y + 3) = 74

2y² + 3y + 9 = 74

2y² + 3y + 9 - 74 = 0

2y² + 3y - 65 = 0

2y² + 13y - 10y - 65 = 0

y(2y + 13) - 5(2y + 13) = 0

y - 5 = 0 or 2y - 13 = 0

y = 5 or y = 13/2

Since an integer is a whole number, then

y = 5

x = y + 3 = 5 + 3

x = 8