Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. The sum of the squares of two consecutive odd integers is 202. Find the integers

Verified step by step guidance

1. Let the first odd integer be represented by \( x \).
2. The next consecutive odd integer can be represented as \( x + 2 \).
3. Write the equation for the sum of the squares of these two integers: \( x^2 + (x + 2)^2 = 202 \).
4. Expand the equation: \( x^2 + (x^2 + 4x + 4) = 202 \).
5. Combine like terms and solve the quadratic equation: \( 2x^2 + 4x + 4 = 202 \).

Key Concepts

Consecutive Odd Integers

Consecutive odd integers are numbers that follow one another in the sequence of odd numbers, such as 1, 3, 5, etc. If 'x' is an odd integer, the next consecutive odd integer can be expressed as 'x + 2'. Understanding this concept is crucial for setting up equations involving odd integers.

Sum of Squares

The sum of squares refers to the operation of squaring numbers and then adding the results together. For example, if 'a' and 'b' are two integers, the sum of their squares is represented as a² + b². In this problem, we need to find the sum of the squares of two consecutive odd integers and set it equal to 202.

Setting Up Equations

Setting up equations involves translating a word problem or mathematical statement into a mathematical expression or equation. In this case, we need to create an equation based on the relationship between the two consecutive odd integers and their sum of squares, which will allow us to solve for the unknown integers.