Sum Of Squares Of Roots Calculation For Equations With Absolute Values

This article delves into the fascinating world of equations and their roots, specifically focusing on calculating the sum of squares of the roots for two distinct equations. We will explore the intricacies of each equation, employing different techniques to solve them and ultimately arrive at the desired sums. This exploration will not only enhance our understanding of algebraic equations but also showcase the power of mathematical problem-solving.

Finding the Sum of Squares of Roots for |x-2|^2 + |x-2| - 2 = 0

Let's begin by tackling the first equation: |x-2|^2 + |x-2| - 2 = 0. This equation involves absolute values, which adds a layer of complexity to the problem. To effectively solve this, we'll employ a substitution method. The key here is recognizing the repeating term |x-2|. By substituting this term, we can transform the equation into a more manageable quadratic form. Let's delve into the step-by-step solution.

First, let's substitute y = |x-2|. This substitution transforms our original equation into a quadratic equation in terms of y: y^2 + y - 2 = 0. This quadratic equation is significantly easier to solve. We can factor this quadratic equation as (y + 2)(y - 1) = 0. This gives us two possible solutions for y: y = -2 and y = 1. Now, the crucial part is to remember that y represents the absolute value |x-2|. Absolute values, by definition, cannot be negative. Therefore, y = -2 is an extraneous solution and we discard it. We are left with y = 1.

Now, we substitute back |x-2| for y, which gives us the equation |x-2| = 1. To solve this absolute value equation, we consider two cases: when the expression inside the absolute value is positive or zero, and when it's negative. In the first case, x - 2 = 1, which leads to x = 3. In the second case, -(x - 2) = 1, which simplifies to -x + 2 = 1, and solving for x gives us x = 1. Thus, the roots of the equation |x-2|^2 + |x-2| - 2 = 0 are x = 1 and x = 3. To find the sum of the squares of these roots, we calculate 1^2 + 3^2 = 1 + 9 = 10. Therefore, the sum of the squares of the roots for the first equation is 10.

This problem highlights the importance of understanding absolute values and the technique of substitution in simplifying equations. By recognizing the structure of the equation and making appropriate substitutions, we transformed a seemingly complex problem into a solvable quadratic equation. The process of checking for extraneous solutions is also crucial when dealing with absolute value equations, ensuring that our final answers are valid. This systematic approach is invaluable in tackling various mathematical problems, fostering a deeper understanding of algebraic principles and problem-solving strategies.

Finding the Sum of Squares of Roots for x^2 - 2|x-3| - 5 = 0

Next, we tackle the second equation: x^2 - 2|x-3| - 5 = 0. This equation, similar to the first, involves an absolute value, but its structure requires a slightly different approach. We will again need to consider different cases based on the sign of the expression inside the absolute value. However, the presence of the x^2 term means we will be dealing with quadratic equations in each case. Let's break down the solution step-by-step.

To solve this equation, we need to consider two cases arising from the absolute value term |x-3|. The first case is when x-3 ≥ 0, which means x ≥ 3. In this case, |x-3| = x-3, and our equation becomes x^2 - 2(x-3) - 5 = 0. Simplifying this equation, we get x^2 - 2x + 6 - 5 = 0, which further simplifies to x^2 - 2x + 1 = 0. This is a perfect square trinomial, which can be factored as (x-1)^2 = 0. Therefore, x = 1 is a potential solution. However, we must remember the condition for this case, which is x ≥ 3. Since 1 is not greater than or equal to 3, this solution is extraneous and must be discarded.

The second case is when x-3 < 0, which means x < 3. In this case, |x-3| = -(x-3), and our equation becomes x^2 - 2(-(x-3)) - 5 = 0. Simplifying this equation, we get x^2 + 2(x-3) - 5 = 0, which further simplifies to x^2 + 2x - 6 - 5 = 0, and finally to x^2 + 2x - 11 = 0. This is a quadratic equation that doesn't factor easily, so we will use the quadratic formula to find the roots. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots are given by x = (-b ± √(b^2 - 4ac)) / (2a). In our case, a = 1, b = 2, and c = -11. Plugging these values into the quadratic formula, we get x = (-2 ± √(2^2 - 4(1)(-11))) / (2(1)), which simplifies to x = (-2 ± √(4 + 44)) / 2, and further to x = (-2 ± √48) / 2. We can simplify √48 as √(16 * 3) = 4√3, so the roots are x = (-2 ± 4√3) / 2, which simplifies to x = -1 ± 2√3.

Now, we need to check if these roots satisfy the condition x < 3. Since √3 is approximately 1.732, 2√3 is approximately 3.464. Thus, -1 + 2√3 is approximately -1 + 3.464 = 2.464, which is less than 3. Also, -1 - 2√3 is approximately -1 - 3.464 = -4.464, which is also less than 3. Therefore, both roots, x = -1 + 2√3 and x = -1 - 2√3, are valid solutions. To find the sum of the squares of these roots, we calculate (-1 + 2√3)^2 + (-1 - 2√3)^2. Expanding these squares, we get (1 - 4√3 + 12) + (1 + 4√3 + 12) = 13 - 4√3 + 13 + 4√3 = 26. Therefore, the sum of the squares of the roots for the second equation is 26.

This problem demonstrates the importance of considering different cases when dealing with absolute value equations. It also highlights the application of the quadratic formula in solving quadratic equations that cannot be easily factored. The process of verifying the solutions against the initial conditions for each case is crucial to ensure the validity of the answers. This methodical approach reinforces the fundamental principles of algebraic problem-solving and enhances our ability to tackle more complex equations.

Summing the Results and Conclusion

Finally, to answer the original question, we need to add the sum of the squares of the roots for both equations. We found that the sum of squares for the first equation is 10, and the sum of squares for the second equation is 26. Adding these together, we get 10 + 26 = 36. Therefore, the sum of the squares of the roots of the equation |x-2|^2 + |x-2| - 2 = 0 and the sum of squares of the roots of the equation x^2 - 2|x-3| - 5 = 0 is 36.

In conclusion, this article has provided a comprehensive exploration of solving equations involving absolute values and calculating the sum of squares of their roots. We have demonstrated the importance of techniques such as substitution, considering different cases based on the absolute value, and applying the quadratic formula. The process of verifying solutions against initial conditions was also emphasized. These skills are essential for mastering algebraic problem-solving and fostering a deeper understanding of mathematical concepts. By systematically breaking down complex problems into manageable steps, we can effectively navigate the intricacies of equations and arrive at accurate solutions. The final answer, 36, represents the culmination of our efforts and showcases the power of mathematical reasoning and problem-solving techniques.