Question

the difference between two positive integers is 8. if the smaller is added to the square of the larger, the sum is 102.
step 2 of 2 : find the integers by solving the equation.

Explanation:

Define variables and set up equations

Using the Integer Word Problems knowledge point
Let \(x\) be the larger positive integer and \(y\) be the smaller positive integer.
The difference between them is 8:

$$x - y = 8 \implies y = x - 8$$

The smaller is added to the square of the larger to get 102:

$$x^2 + y = 102$$

Substitute and form a quadratic equation

Using the Quadratic Word Problems knowledge point
Substitute \(y = x - 8\) into the second equation:

$$x^2 + (x - 8) = 102$$

$$x^2 + x - 110 = 0$$

Solve the quadratic equation

Using the Solving Quadratic Equations knowledge point

$$
LATEXBLOCK0
$$

Determine the positive integer values

Since the problem states both are positive integers, we choose:

$$x = 10$$

Find the smaller integer \(y\):

$$y = 10 - 8 = 2$$

The two positive integers are 2 and 10.

Answer:

The difference between two positive integers is 8. If the smaller is added to the square of the larger, the sum is 102.
Step 2 of 2 : Find the integers by solving the equation.

Answer
<blank>2</blank> and <blank>10</blank>