Question

the difference between two positive integers is 4. if the smaller is added to the square of the larger, the sum is 52.

step 2 of 2 : find the integers by solving the equation.

Explanation:

Define the variables and set up the system

Using the Translating Word Problems to Quadratic Equations knowledge point:
Let \(x\) be the smaller positive integer and \(y\) be the larger positive integer.
The difference between the two positive integers is 4:

$$ y - x = 4 \implies y = x + 4 $$

The smaller is added to the square of the larger, and the sum is 52:

$$ x + y^2 = 52 $$

Substitute and form a quadratic equation

Using the Translating Word Problems to Quadratic Equations knowledge point:
Substitute \(y = x + 4\) into the second equation:

$$ x + (x + 4)^2 = 52 $$

Expand and simplify:

$$
LATEXBLOCK0
$$

Solve the quadratic equation by factoring

Using the Factoring Quadratic Equations and Solving Quadratic Equations knowledge points:

$$
LATEXBLOCK1
$$

Determine the positive integer solutions

Since the problem states that both integers must be positive, we discard the negative solution:

$$
LATEXBLOCK2
$$

The two positive integers are 3 and 7.

Answer:

The integers are 3 and 7.