The Sum Of The Squares Of Two Consecutive Even Real Numbers Is 52. Find The Numbers.
Introduction
In the realm of mathematics, particularly in algebra, we often encounter problems that require us to find unknown numbers based on given conditions. One such problem involves the sum of squares of two consecutive even real numbers. These types of problems not only test our algebraic skills but also our ability to translate word problems into mathematical equations. Let's delve into a specific instance of this problem: finding two consecutive even real numbers whose squares add up to 52. This article will explore the steps involved in solving this problem, offering a comprehensive guide for students and math enthusiasts alike. We will break down the problem, establish the equation, solve for the unknowns, and verify our solution. This methodical approach will help you tackle similar problems with confidence.
Understanding the Problem
Before we jump into the solution, it's crucial to understand the core concepts involved. The problem states that we need to find two numbers. These numbers are not just any numbers; they are consecutive even real numbers. Real numbers encompass all numbers that can be represented on a number line, including integers, rational numbers, and irrational numbers. Even numbers are integers that are divisible by 2, such as -4, -2, 0, 2, 4, and so on. Consecutive even numbers are even numbers that follow each other in sequence, with a difference of 2 between them. For example, 2 and 4, or -6 and -4, are consecutive even numbers.
The problem also mentions the "sum of squares." This means we need to square each of the two numbers (multiply the number by itself) and then add the results together. The final piece of information is that this sum equals 52. This gives us a clear target to aim for. By understanding these key concepts, we can translate the word problem into a mathematical equation and proceed with solving it. Identifying the type of numbers we are looking for (consecutive even real numbers) and understanding the mathematical operations involved (squaring and summing) are essential first steps in tackling this problem.
Setting Up the Equation
Now that we understand the problem, the next step is to translate the words into a mathematical equation. This is a crucial step in solving any word problem. Let's represent the two consecutive even real numbers using algebraic variables. Let x be the first even number. Since the numbers are consecutive and even, the next even number will be x + 2. For example, if x is 4, then x + 2 would be 6, which is the next consecutive even number.
The problem states that the sum of the squares of these two numbers is 52. This can be written as an equation: x² + (x + 2)² = 52. Here, x² represents the square of the first even number, and (x + 2)² represents the square of the second even number. The "+" sign indicates that we are adding these squares together, and the "= 52" completes the equation by stating that the sum is equal to 52. This equation now encapsulates all the information given in the problem statement. By setting up the equation correctly, we have transformed the word problem into a mathematical form that we can solve using algebraic techniques. The accurate translation into an algebraic equation is the foundation for finding the solution.
Solving the Equation
With the equation x² + (x + 2)² = 52 in hand, we can now proceed to solve for x. The first step is to expand the equation by squaring the term (x + 2). Recall that (a + b)² = a² + 2ab + b². Applying this to our equation, we get (x + 2)² = x² + 4x + 4. Substituting this back into the original equation, we have x² + (x² + 4x + 4) = 52.
Next, we simplify the equation by combining like terms. We have two x² terms, so we can combine them to get 2x². The equation now looks like this: 2x² + 4x + 4 = 52. To solve a quadratic equation, it's best to set it equal to zero. Subtracting 52 from both sides, we get 2x² + 4x - 48 = 0. Now we have a standard quadratic equation in the form ax² + bx + c = 0.
We can simplify the equation further by dividing all terms by 2, which gives us x² + 2x - 24 = 0. This simplifies the coefficients and makes the equation easier to factor. Now we need to factor the quadratic expression. We are looking for two numbers that multiply to -24 and add to 2. These numbers are 6 and -4. Therefore, we can factor the equation as (x + 6)(x - 4) = 0. Setting each factor equal to zero gives us two possible solutions for x: x + 6 = 0 or x - 4 = 0. Solving these equations, we find x = -6 or x = 4. These are the two possible values for the first even number. By systematically expanding, simplifying, and factoring the equation, we have arrived at the potential solutions for x.
Finding the Two Numbers
Now that we have found the possible values for x, we need to determine the two consecutive even numbers that satisfy the original problem. Recall that x represents the first even number, and x + 2 represents the second even number. We found two possible values for x: -6 and 4. Let's consider each case.
Case 1: If x = -6, then the first even number is -6. The second even number is x + 2 = -6 + 2 = -4. So, one pair of consecutive even numbers is -6 and -4.
Case 2: If x = 4, then the first even number is 4. The second even number is x + 2 = 4 + 2 = 6. So, another pair of consecutive even numbers is 4 and 6. We now have two pairs of numbers that could potentially satisfy the problem's conditions. It's important to check these solutions to ensure they are correct. By substituting the values of x back into the expressions for the two numbers, we have identified two possible pairs of consecutive even numbers.
Verification
The final step in solving any mathematical problem is verification. We need to check if the pairs of numbers we found actually satisfy the given condition: that the sum of their squares is 52. Let's check each pair.
For the pair -6 and -4, we need to calculate (-6)² + (-4)². (-6)² is (-6) * (-6) = 36, and (-4)² is (-4) * (-4) = 16. Adding these together, we get 36 + 16 = 52. So, the pair -6 and -4 satisfies the condition.
For the pair 4 and 6, we need to calculate 4² + 6². 4² is 4 * 4 = 16, and 6² is 6 * 6 = 36. Adding these together, we get 16 + 36 = 52. So, the pair 4 and 6 also satisfies the condition.
Both pairs of numbers, -6 and -4, and 4 and 6, satisfy the given condition. Therefore, these are the solutions to the problem. By checking our solutions against the original problem statement, we have ensured the accuracy of our results.
Conclusion
In this article, we successfully solved the problem of finding two consecutive even real numbers whose squares add up to 52. We started by understanding the problem and identifying the key concepts involved, such as consecutive even numbers and the sum of squares. Then, we translated the word problem into a mathematical equation: x² + (x + 2)² = 52. Next, we solved the equation by expanding, simplifying, and factoring it, which gave us two possible values for x: -6 and 4. Using these values, we found two pairs of numbers: -6 and -4, and 4 and 6. Finally, we verified our solutions by checking if the sum of the squares of each pair equaled 52, which they did.
This problem demonstrates the importance of a systematic approach to solving mathematical problems. By breaking down the problem into smaller steps, setting up the equation correctly, solving it methodically, and verifying the solutions, we can confidently tackle even complex problems. This approach is not only useful in mathematics but also in many other areas of life where problem-solving skills are essential. Remember, practice makes perfect, so the more you work through problems like this, the more proficient you will become at solving them. Understanding the underlying principles and applying them step-by-step is the key to success in mathematics and beyond.