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 224.

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Let the first even integer be represented by \(x\). Since \(x\) is an even integer, the next consecutive even integer can be represented as \(x + 2\).

According to the problem, the product of these two consecutive even integers is 224. So, we can write the equation: \(x \times (x + 2) = 224\).

Expand the left side of the equation to get a quadratic equation: \(x^2 + 2x = 224\).

Bring all terms to one side to set the equation equal to zero: \(x^2 + 2x - 224 = 0\).

Solve the quadratic equation \(x^2 + 2x - 224 = 0\) 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 \(x\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 sequences of even numbers.

Algebraic Representation of Unknowns

Representing unknown numbers with variables allows us to translate word problems into algebraic equations. Here, if x is the first even integer, then x + 2 is the next consecutive even integer. This representation is essential for forming equations to solve for unknown values.

Solving Quadratic Equations

When the product of two integers is given, setting up an equation often leads to a quadratic equation. Solving quadratic equations involves rearranging terms, factoring, or using the quadratic formula to find the values of the variable. This step is crucial to determine the integers that satisfy the problem.