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. Find two consecutive even integers whose product is 168.

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 product of these two consecutive even integers is 168. So, we can write the equation: \(x \times (x + 2) = 168\).
3. Expand the left side of the equation to get a quadratic equation: \(x^2 + 2x = 168\).
4. Bring all terms to one side to set the equation equal to zero: \(x^2 + 2x - 168 = 0\).
5. Solve the quadratic equation \(x^2 + 2x - 168 = 0\) using factoring, completing the square, or the quadratic formula to find the values of \(x\), which represent the first even integer. Then find the second integer by adding 2 to \(x\).

Key Concepts

Consecutive Even Integers

Consecutive even integers are even numbers that follow one another in order, each differing by 2. For example, if x is an even integer, then the next consecutive even integer is x + 2. Understanding this helps in setting up expressions for problems involving consecutive even numbers.

Forming Algebraic Expressions from Word Problems

This involves translating a real-world situation into algebraic terms. Here, representing the two consecutive even integers as x and x + 2 allows us to create an equation based on their product, which is given as 168. This step is crucial for solving the problem using algebra.

Solving Quadratic Equations

When the product of two expressions is given, setting up an equation often leads to a quadratic equation. Solving it involves rearranging terms, factoring or using the quadratic formula to find the integer values of x that satisfy the equation, which represent the consecutive even integers.