Solving The Quadratic Equation X² + 2x = -2 A Step By Step Guide

In the vast landscape of mathematics, quadratic equations stand as fundamental pillars, shaping our understanding of curves, trajectories, and countless real-world phenomena. Quadratic equations, characterized by their second-degree polynomial form, present a unique challenge and beauty in their solutions. Our quest today involves unraveling the solution to a specific quadratic equation: x² + 2x = -2. This exploration will not only demonstrate the method for finding solutions but also illuminate the underlying principles that govern these equations. To solve a quadratic equation, we often aim to bring it into the standard form of ax² + bx + c = 0, where a, b, and c are constants. This form allows us to employ various techniques such as factoring, completing the square, or the quadratic formula. Each method offers a unique pathway to the solution, and the choice often depends on the specific characteristics of the equation at hand. The beauty of mathematics lies in the multiple avenues that lead to the same destination – the solution. In this particular case, we will delve into the process of transforming the given equation into its standard form and then strategically apply a method that reveals its hidden solutions. The journey through this equation will not only provide a numerical answer but also deepen our appreciation for the elegance and power of algebraic manipulation. As we embark on this mathematical expedition, let's keep in mind that each step is a deliberate stride towards unveiling the solution, a testament to the logical and consistent nature of mathematics.

The first crucial step in solving the quadratic equation x² + 2x = -2 is to transform it into the standard form. This form, ax² + bx + c = 0, is the key to unlocking the equation's secrets. By rearranging the terms, we can create a clear pathway to apply methods like the quadratic formula or completing the square. Our initial equation, x² + 2x = -2, presents a slight obstacle in its current state. The constant term, -2, resides on the right side of the equation, hindering our progress towards the standard form. To rectify this, we employ a fundamental principle of algebra: maintaining balance. We add 2 to both sides of the equation, ensuring that the equality remains intact. This seemingly simple operation is a cornerstone of algebraic manipulation, allowing us to rearrange terms without disrupting the equation's inherent truth. The result of this addition is a transformed equation: x² + 2x + 2 = 0. Now, the equation proudly stands in its standard form, ready for the next stage of our solution-seeking journey. The coefficients are now clearly visible: a = 1, b = 2, and c = 2. These coefficients hold the key to applying the quadratic formula, a powerful tool that can solve any quadratic equation, regardless of its complexity. As we move forward, remember that this transformation is not merely a cosmetic change. It is a strategic maneuver that unveils the equation's underlying structure, making it amenable to established solution techniques. The standard form serves as a universal language for quadratic equations, allowing mathematicians to communicate and collaborate effectively in the pursuit of solutions.

With the equation now in its standard form, x² + 2x + 2 = 0, we introduce the quadratic formula, a powerful tool capable of solving any quadratic equation. This formula, often hailed as a universal solver, provides a direct pathway to the solutions, regardless of the equation's complexity. The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x are given by: x = (-b ± √(b² - 4ac)) / 2a. This elegant expression encapsulates the relationship between the coefficients of the equation and its solutions. In our case, we have identified a = 1, b = 2, and c = 2. These values are the coordinates that guide us through the formula, leading us to the equation's hidden solutions. Substituting these values into the quadratic formula, we get: x = (-2 ± √(2² - 4 * 1 * 2)) / (2 * 1). This substitution is a critical step, transforming the abstract formula into a concrete calculation tailored to our specific equation. The next step involves simplifying the expression, carefully navigating the arithmetic operations to arrive at the solutions. The discriminant, b² - 4ac, plays a pivotal role in determining the nature of the solutions. If the discriminant is positive, we have two distinct real solutions. If it is zero, we have one real solution (a repeated root). And if, as in our case, it is negative, we encounter complex solutions, numbers that extend beyond the realm of the real number line. The quadratic formula is not just a formula; it is a testament to the power of mathematical abstraction. It provides a generalized solution that applies to an infinite family of equations, a remarkable feat of mathematical ingenuity. As we continue our calculation, we will witness the formula in action, transforming the equation's coefficients into the elusive solutions.

As we delve deeper into the quadratic formula, x = (-2 ± √(2² - 4 * 1 * 2)) / (2 * 1), we encounter a fascinating twist – complex solutions. The presence of a negative value under the square root signals the emergence of imaginary numbers, expanding our mathematical horizon beyond the real number line. Simplifying the expression under the square root, we get: 2² - 4 * 1 * 2 = 4 - 8 = -4. The negative discriminant, -4, confirms our suspicion of complex solutions. The square root of -4 is expressed as 2i, where 'i' is the imaginary unit, defined as the square root of -1. This introduction of 'i' allows us to express the solutions in terms of complex numbers, which have both a real and an imaginary component. Substituting 2i back into the quadratic formula, we have: x = (-2 ± 2i) / 2. Now, we can simplify this expression by dividing both the real and imaginary parts by 2, yielding: x = -1 ± i. This elegant result reveals two complex solutions: x = -1 + i and x = -1 - i. These solutions are not mere abstract entities; they have profound implications in various fields, including electrical engineering, quantum mechanics, and signal processing. Complex numbers, initially conceived as a mathematical curiosity, have proven to be indispensable tools for understanding and modeling the world around us. Our journey through this quadratic equation has not only provided us with numerical answers but has also exposed us to the beauty and power of complex numbers. The solutions, x = -1 + i and x = -1 - i, are a testament to the completeness of the complex number system, which allows us to solve equations that have no solutions in the realm of real numbers. As we reflect on this exploration, we appreciate the interconnectedness of mathematical concepts, where seemingly disparate ideas converge to provide a deeper understanding of the mathematical landscape.

With the solutions x = -1 + i and x = -1 - i in hand, the final step in our mathematical journey is to verify their correctness. This crucial step ensures that our calculations are accurate and that the solutions indeed satisfy the original equation, x² + 2x = -2. To verify a solution, we substitute it back into the original equation and check if the equality holds. Let's start with x = -1 + i. Substituting this value into the equation, we get: (-1 + i)² + 2(-1 + i) = -2. Expanding the terms, we have: (1 - 2i + i²) + (-2 + 2i) = -2. Recall that i² = -1, so we can further simplify the expression: (1 - 2i - 1) + (-2 + 2i) = -2. Combining the real and imaginary terms, we get: -2i - 2 + 2i = -2. The imaginary terms cancel out, leaving us with: -2 = -2. This equality confirms that x = -1 + i is indeed a valid solution to the equation. Now, let's verify the second solution, x = -1 - i. Substituting this value into the equation, we get: (-1 - i)² + 2(-1 - i) = -2. Expanding the terms, we have: (1 + 2i + i²) + (-2 - 2i) = -2. Again, using i² = -1, we simplify the expression: (1 + 2i - 1) + (-2 - 2i) = -2. Combining the real and imaginary terms, we get: 2i - 2 - 2i = -2. The imaginary terms cancel out, leaving us with: -2 = -2. This equality confirms that x = -1 - i is also a valid solution. The successful verification of both solutions reinforces our confidence in the accuracy of our calculations and the power of the quadratic formula. This final step is not just a formality; it is a testament to the rigor and precision that define mathematics. As we conclude our exploration, we appreciate the journey that has taken us from the initial equation to the verified solutions, a testament to the beauty and consistency of mathematical reasoning.

In conclusion, our mathematical exploration of the quadratic equation x² + 2x = -2 has been a resounding success. We have navigated the intricacies of quadratic equations, transformed the given equation into its standard form, and skillfully applied the quadratic formula to unveil its solutions. The journey has not only provided us with numerical answers but has also deepened our understanding of the underlying principles that govern these equations. We have encountered the fascinating realm of complex numbers, witnessing their emergence as solutions to equations that have no real roots. The solutions, x = -1 + i and x = -1 - i, stand as a testament to the completeness of the complex number system and its ability to solve a wider range of problems than the real number system alone. The verification process has further solidified our confidence in the accuracy of our calculations, reinforcing the importance of rigor and precision in mathematics. This exploration is not just about solving a single equation; it is about developing a deeper appreciation for the power and beauty of mathematical reasoning. The techniques and concepts we have encountered are applicable to a vast array of problems, extending far beyond the specific equation we have addressed. As we reflect on this journey, we recognize the interconnectedness of mathematical ideas, where concepts build upon each other to create a rich and intricate tapestry of knowledge. The ability to solve quadratic equations is a fundamental skill in mathematics, and it serves as a stepping stone to more advanced topics. Our exploration has not only equipped us with this skill but has also instilled in us a sense of curiosity and a desire to delve deeper into the world of mathematics. The triumph of this mathematical exploration lies not just in the solutions we have found but also in the intellectual growth we have experienced along the way.