Solving the Equation for Two Consecutive Even Numbers with Sum of Squares 100

This article delves into a classic algebra problem: finding two consecutive even numbers such that the sum of their squares is 100. We explore different methods to solve this problem, including a straightforward approach and a step-by-step algebraic solution, and discuss the significance of each method.

Introduction

Algebraic equations often present interesting challenges, and the problem of finding two consecutive even numbers whose squares sum up to a given number (in this case, 100) is no exception. While some approaches involve complex algebraic manipulations, we'll demonstrate a simpler method through both direct reasoning and algebraic steps.

Simple Reasoning Method

First, let's consider a simpler and more intuitive approach. If we denote the two consecutive even numbers as x and x 2, their squares must be close to half of the target sum, 100. By dividing 100 by 2, we find that 50 is just above the square of 7, which is 49. Therefore, the two numbers that fit this description are likely to be 6 and 8:

(quad 6^2 8^2 36 64 100)

This method is straightforward and quick, especially when dealing with simpler problems.

Algebraic Solution

For a more rigorous approach, we can set up and solve a quadratic equation. We start by assuming the two consecutive even numbers to be x and x 2, and their squares to sum to 100:

(quad x^2 (x 2)^2 100)

Expanding and simplifying the equation:

(quad x^2 (x^2 4x 4) 100)

(quad 2x^2 4x 4 100)

(quad 2x^2 4x 4 - 100 0)

(quad 2x^2 4x - 96 0)

Dividing the entire equation by 2:

(quad x^2 2x - 48 0)

Factoring the quadratic equation:

(quad (x 8)(x - 6) 0)

Setting each factor to zero gives us the solutions:

(quad x 8 0 quad rightarrow quad x -8) or (quad x - 6 0 quad rightarrow quad x 6)

Therefore, the two pairs of consecutive even numbers are:

(quad -8 text{ and } -6) or (quad 6 text{ and } 8)

In summary, the two consecutive even numbers whose squares sum to 100 are 6 and 8 or -8 and -6, depending on the context.

Conclusion

This problem highlights the importance of both intuitive reasoning and algebraic techniques in solving mathematical challenges. While the simple reasoning method provides a quick insight, the algebraic approach ensures a solid understanding and can be applied to more complex problems.

Keywords: consecutive even numbers, quadratic equation, sum of squares