Use 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 \(x\). Since \(x\) is an odd integer, the next consecutive odd integer can be represented as \(x + 2\).
2. Write an expression for the sum of the squares of these two consecutive odd integers: \(x^2 + (x + 2)^2\).
3. Set up the equation based on the problem statement: \(x^2 + (x + 2)^2 = 202\).
4. Expand the squared term: \(x^2 + (x^2 + 4x + 4) = 202\).
5. Combine like terms to form a quadratic equation: \$2x^2 + 4x + 4 = 202$. Then, subtract 202 from both sides to set the equation to zero.

Key Concepts

Consecutive Odd Integers

Consecutive odd integers are odd numbers that follow one another in order, each differing by 2. If x is an odd integer, then the next consecutive odd integer is x + 2. This concept helps in expressing the two integers algebraically for problem-solving.

Algebraic Representation of Word Problems

Translating word problems into algebraic expressions involves defining variables to represent unknowns and writing equations based on given relationships. Here, representing the two consecutive odd integers as x and x + 2 allows setting up an equation involving their squares.

Solving Quadratic Equations

When the sum of squares is given, the resulting equation is quadratic. Solving quadratic equations involves rearranging terms, factoring or using the quadratic formula, and interpreting solutions in the context of the problem to find integer values.