Solving For Consecutive Even Integers With A Product Of 24

Understanding the Problem

In this mathematical puzzle, we are tasked with finding pairs of consecutive even integers whose product equals twenty-four. To effectively address this problem, we must first define what consecutive even integers are and then formulate a strategy for identifying the pairs that satisfy the given condition. Consecutive even integers are even numbers that follow each other in sequence, such as 2 and 4, 10 and 12, or -6 and -4. The key to solving this lies in recognizing the algebraic relationship between these numbers and then using that relationship to solve for the unknown integers. The challenge presented here serves as a valuable exercise in algebraic problem-solving and enhances our understanding of number properties.

To approach this problem methodically, we will translate the word problem into an algebraic equation. Let's represent the first even integer as "x". Since we are looking for consecutive even integers, the next even integer will be "x + 2". The problem states that the product of these two integers is twenty-four. Therefore, we can write the equation as: x(x + 2) = 24. This equation forms the foundation of our solution. By solving this quadratic equation, we will find the possible values of "x", which will then lead us to the pairs of consecutive even integers that meet the given condition. It’s important to remember that quadratic equations can have multiple solutions, so we need to consider both positive and negative possibilities for "x". This careful and systematic approach ensures that we identify all valid pairs of integers.

Solving this type of problem requires a blend of algebraic manipulation and logical reasoning. First, we need to expand the equation and rearrange it into a standard quadratic form, which is ax² + bx + c = 0. This form allows us to easily apply various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Once we have the solutions for "x", we must verify that these solutions indeed give us consecutive even integers and that their product is twenty-four. This verification step is crucial to ensure the accuracy of our results. Furthermore, we should consider both positive and negative solutions, as the problem does not explicitly restrict the integers to be positive. By meticulously working through these steps, we can confidently arrive at the correct answer(s) and deepen our understanding of problem-solving strategies in mathematics.

Setting Up the Equation

To solve the problem of finding two consecutive even integers whose product is 24, we begin by translating the word problem into a mathematical equation. This algebraic representation is the cornerstone of our solution process. Let's denote the first even integer as "x". Since we are seeking consecutive even integers, the next even integer will be "x + 2". This is because even integers are always separated by a difference of 2 (e.g., 2, 4, 6, 8...). The problem explicitly states that the product of these two consecutive even integers is 24. Therefore, we can express this relationship as an equation:

x(x + 2) = 24

This equation is a quadratic equation, which means it involves a variable raised to the power of 2. Solving quadratic equations typically involves rearranging the equation into a standard form and then applying methods such as factoring, completing the square, or using the quadratic formula. The equation we have set up captures the essence of the problem and allows us to use algebraic techniques to find the values of "x" that satisfy the given condition. The accuracy of our solution hinges on the correct interpretation of the problem and the accurate translation into this algebraic form. From here, we will manipulate this equation to find the possible values of "x", which will lead us to the consecutive even integers we are looking for.

Now that we have the equation x(x + 2) = 24, the next step is to expand and simplify it into a standard quadratic form. Expanding the left side of the equation, we get: x² + 2x = 24. To transform this into the standard quadratic form, we need to move all terms to one side of the equation, leaving zero on the other side. This is achieved by subtracting 24 from both sides of the equation, which gives us: x² + 2x - 24 = 0. This equation is now in the standard quadratic form ax² + bx + c = 0, where a = 1, b = 2, and c = -24. Having the equation in this form is crucial because it allows us to easily apply various methods for solving quadratic equations. The standard form provides a clear structure for identifying the coefficients, which are essential for methods like factoring and the quadratic formula. This transformation is a critical step in solving the problem, as it sets the stage for the algebraic techniques we will use to find the solutions for "x".

Solving the Quadratic Equation

With the equation in the standard quadratic form (x² + 2x - 24 = 0), we can now proceed to solve for "x". There are several methods available for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a particularly efficient method. Factoring involves expressing the quadratic expression as a product of two binomials. To factor the quadratic expression x² + 2x - 24, we need to find two numbers that multiply to -24 and add up to 2. These two numbers are 6 and -4. Therefore, we can factor the quadratic expression as:

(x + 6)(x - 4) = 0

This factored form of the equation is crucial because it allows us to easily find the values of "x" that make the equation true. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This means that either (x + 6) = 0 or (x - 4) = 0. Solving these two linear equations will give us the possible values of "x". This factoring technique is a fundamental tool in algebra and is particularly useful for solving quadratic equations when the coefficients are integers and the roots are rational. The ability to factor quadratic expressions efficiently is a valuable skill in mathematical problem-solving.

Now that we have the factored form of the equation, (x + 6)(x - 4) = 0, we can find the solutions for "x". Setting each factor equal to zero, we get two equations:

1. x + 6 = 0
2. x - 4 = 0

Solving the first equation, x + 6 = 0, we subtract 6 from both sides to isolate "x", which gives us x = -6. Solving the second equation, x - 4 = 0, we add 4 to both sides to isolate "x", which gives us x = 4. Therefore, the two possible values for "x" are -6 and 4. These values represent the first even integer in our pairs of consecutive even integers. It is important to remember that quadratic equations can have up to two distinct real solutions, and in this case, we have found both. These solutions are the foundation for identifying the pairs of consecutive even integers that satisfy the original problem statement. We now need to use these values of "x" to find the corresponding consecutive even integers and verify that their product is indeed 24. This step ensures that our solutions are accurate and complete.

Identifying the Integer Pairs

Now that we have found the two possible values for "x" (-6 and 4), we can identify the pairs of consecutive even integers. Recall that "x" represents the first even integer, and the next consecutive even integer is "x + 2". Let's consider each value of "x" separately.

First, let's take x = -6. The next consecutive even integer would be -6 + 2 = -4. So, one pair of consecutive even integers is -6 and -4. To verify that this pair satisfies the problem's condition, we multiply the two integers: (-6) * (-4) = 24. This confirms that -6 and -4 is a valid pair.

Next, let's consider x = 4. The next consecutive even integer would be 4 + 2 = 6. So, another pair of consecutive even integers is 4 and 6. To verify this pair, we multiply the two integers: 4 * 6 = 24. This also confirms that 4 and 6 is a valid pair.

Therefore, we have identified two pairs of consecutive even integers whose product is 24: -6 and -4, and 4 and 6. These pairs are the solutions to the problem. It is crucial to verify each pair to ensure that they meet the given condition, as this confirms the accuracy of our algebraic solution. This step-by-step approach ensures that we have identified all possible solutions and have not overlooked any valid pairs. The process of finding these pairs demonstrates the practical application of solving quadratic equations in real-world scenarios.

Final Answer

In summary, we have successfully solved the problem of finding two consecutive even integers whose product is twenty-four. By translating the problem into an algebraic equation, solving the resulting quadratic equation, and verifying the solutions, we have identified two pairs of integers that satisfy the given condition. The two pairs of consecutive even integers are:

1. -6 and -4
2. 4 and 6

These are the final answers to the problem. The process involved setting up the equation x(x + 2) = 24, expanding and rearranging it into the standard quadratic form x² + 2x - 24 = 0, factoring the quadratic expression into (x + 6)(x - 4) = 0, and solving for "x" to obtain the values -6 and 4. We then used these values to find the corresponding consecutive even integers and verified that their products equal 24. This methodical approach highlights the importance of translating word problems into mathematical expressions and utilizing algebraic techniques to find solutions. The problem showcases how quadratic equations can model real-world scenarios and how solving them can lead to meaningful results. The solutions -6 and -4, and 4 and 6, demonstrate that mathematical problems can sometimes have multiple correct answers, each of which satisfies the given conditions.