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 1 of 2: set up an equation to solve the given word problem.

answer
\\(x^2 + x + 8 = 102\\)
\\(x + x + 8 = 102\\)
\\(x + (x + 8)^2 = 102\\)
\\(x(x + 8)^2 = 102\\)

Explanation:

Define variables for the integers

Using the Integer Word Problems knowledge point
Let \(x\) represent the smaller positive integer.
Since the difference between the two positive integers is 8, the larger integer is:

$$ x + 8 $$

Translate the relationship into an equation

Using the Quadratic Word Problems knowledge point
The problem states: "If the smaller is added to the square of the larger, the sum is 102."

• Smaller integer: \(x\)
• Square of the larger integer: \((x + 8)^2\)
• Sum: \(x + (x + 8)^2 = 102\)

Answer:

• (A) \(x^2 + x + 8 = 102\)
• (B) \(x + x + 8 = 102\)
• (C) \(x + (x + 8)^2 = 102\) (Correct answer)
• (D) \(x(x + 8)^2 = 102\)