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 even integers is 52. Find the integers.

Verified step by step guidance

1. Let the first even integer be represented by \(x\). Since \(x\) is an even integer, the next consecutive even integer can be represented as \(x + 2\).
2. According to the problem, the sum of the squares of these two consecutive even integers is 52. This can be written as the equation: \(x^2 + (x + 2)^2 = 52\).
3. Expand the squared term \((x + 2)^2\) using the formula \((a + b)^2 = a^2 + 2ab + b^2\), which gives \(x^2 + 4x + 4\).
4. Substitute the expanded form back into the equation to get: \(x^2 + x^2 + 4x + 4 = 52\).
5. Combine like terms to form a quadratic equation: \$2x^2 + 4x + 4 = 52\(. Then, subtract 52 from both sides to set the equation to zero: \)2x^2 + 4x + 4 - 52 = 0\(, which simplifies to \)2x^2 + 4x - 48 = 0$.

Key Concepts

Consecutive Even Integers

Consecutive even integers are even numbers that follow one another in order, each differing by 2. If x is an even integer, then the next consecutive even integer is x + 2. This concept helps in setting up expressions for problems involving sequences of even numbers.

Algebraic Representation of Word Problems

Translating word problems into algebraic expressions involves defining variables to represent unknowns and writing equations based on the problem's conditions. For example, representing two consecutive even integers as x and x + 2 allows us to form equations to solve for x.

Solving Quadratic Equations

Quadratic equations involve variables raised to the second power and can be solved by factoring, completing the square, or using the quadratic formula. In this problem, the sum of squares leads to a quadratic equation, which must be solved to find the integer values.