Solving The Equation X^2 - 7 = 0

Understanding the Equation

In this discussion, we will be focusing on solving the quadratic equation ${ x^2 - 7 = 0 }$. This equation falls under the category of mathematics, specifically algebra, and it's crucial to grasp the fundamental principles to arrive at the correct solution. Before diving into the step-by-step process, let's break down the components of the equation. The term ${ x^2 }$ represents the variable ${ x }$ raised to the power of 2, which is a key characteristic of quadratic equations. The constant term is -7, and our objective is to find the value(s) of ${ x }$ that satisfy the equation, making the left-hand side equal to zero. This involves isolating ${ x }$ and determining its possible values. To effectively tackle this problem, we'll employ algebraic techniques such as isolating the variable term and applying the square root property. Understanding the structure and properties of quadratic equations is essential for solving not just this specific problem but a wide range of mathematical challenges. The process involves manipulating the equation to get ${ x^2 }$ by itself on one side and then taking the square root of both sides to find ${ x }$. This method leverages the inverse relationship between squaring and taking the square root, allowing us to systematically solve for the unknown variable. By meticulously following these steps, we can confidently determine the values of ${ x }$ that make the equation true. Moreover, this exercise reinforces essential algebraic skills and provides a foundation for more advanced mathematical concepts. The ability to solve quadratic equations is a cornerstone of algebra and has applications in various fields, including physics, engineering, and computer science. Therefore, mastering this skill is invaluable for anyone pursuing studies or careers in these areas. In addition to understanding the algebraic manipulations involved, it's also important to recognize the nature of the solutions. Quadratic equations can have two real solutions, one real solution (a repeated root), or two complex solutions. In this particular case, we expect to find two distinct real solutions because the constant term is negative, which indicates that the graph of the equation (a parabola) will intersect the x-axis at two points. The solutions we obtain will represent the x-coordinates of these intersection points, further highlighting the connection between algebra and geometry. This understanding is crucial for visualizing the problem and interpreting the solutions in a broader context.

Step-by-Step Solution

To solve the equation ${ x^2 - 7 = 0 }$, we follow a straightforward algebraic process. First, we need to isolate the ${ x^2 }$ term. This can be achieved by adding 7 to both sides of the equation. This maintains the equality and moves the constant term to the right-hand side, simplifying the equation for further manipulation. The addition property of equality is a fundamental principle in algebra, ensuring that any operation performed on one side of the equation is also performed on the other side, thereby preserving the balance. By adding 7 to both sides, we transform the equation into ${ x^2 = 7 }$. Now, we have a much simpler equation where the variable term is isolated. The next step involves undoing the square. To do this, we take the square root of both sides of the equation. This is based on the principle that the square root is the inverse operation of squaring. However, it's crucial to remember that when taking the square root, we must consider both positive and negative roots. This is because both a positive number and its negative counterpart, when squared, will yield the same positive result. Therefore, taking the square root of ${ x^2 }$ gives us two possibilities: ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$. The square root of 7 is an irrational number, meaning it cannot be expressed as a simple fraction. It's approximately equal to 2.646, but for exact solutions, we leave it in radical form. Thus, the two solutions to the equation are ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$. These values represent the points where the parabola defined by the equation ${ y = x^2 - 7 }$ intersects the x-axis. Understanding the dual nature of square roots is essential for solving quadratic equations accurately. Failing to consider both positive and negative roots can lead to incomplete or incorrect solutions. This step highlights the importance of paying attention to the properties of mathematical operations and their inverses. Moreover, this process demonstrates the power of algebraic manipulation in simplifying complex problems into manageable steps. By breaking down the equation and applying appropriate operations, we can systematically arrive at the solutions. The solutions ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$ are the precise values that satisfy the original equation, making the left-hand side equal to zero. This confirms the validity of our solution process and underscores the fundamental principles of algebra. The ability to solve such equations is a critical skill in mathematics and has wide-ranging applications in various scientific and engineering disciplines.

The Solutions

In summary, after solving the quadratic equation ${ x^2 - 7 = 0 }$, we have found two solutions for ${ x }$. These solutions are ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$. It is crucial to understand the significance of having two solutions in this context. Quadratic equations, which are equations of the form ${ ax^2 + bx + c = 0 }$, typically have two solutions due to the nature of the squared term. These solutions represent the points where the graph of the quadratic function intersects the x-axis, also known as the roots or zeros of the equation. In our specific case, the graph of ${ y = x^2 - 7 }$ is a parabola that opens upwards. The solutions ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$ correspond to the x-coordinates of the two points where this parabola intersects the x-axis. These intersections are symmetrically located about the y-axis, which is a characteristic of quadratic equations where the linear term (${ bx }$) is absent. The positive solution, ${ x = \sqrt{7} }$, is the x-coordinate of the point where the parabola intersects the x-axis on the positive side, while the negative solution, ${ x = -\sqrt{7} }$, corresponds to the intersection on the negative side. These solutions are irrational numbers, meaning they cannot be expressed as a simple fraction. They are represented exactly using the square root symbol, ${ \sqrt{} }$, which denotes the principal (positive) square root. Approximating these values to a decimal representation would introduce rounding errors, so it's generally preferred to leave them in their exact radical form for accuracy. Understanding the nature and interpretation of solutions is a fundamental aspect of solving mathematical problems. It's not just about finding numerical answers; it's also about understanding what those answers represent in the context of the problem. In this case, the solutions ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$ provide a complete understanding of the equation ${ x^2 - 7 = 0 }$. They not only satisfy the equation but also provide valuable insights into the graphical representation of the corresponding quadratic function. Moreover, this exercise reinforces the importance of considering both positive and negative roots when solving equations involving square roots. This is a common source of error, and recognizing the dual nature of square roots is crucial for accurate solutions. The solutions we obtained are not just abstract numbers; they are concrete values that have specific meaning within the context of the equation and its graphical representation. This deeper understanding is essential for applying mathematical concepts to real-world problems and for further studies in mathematics and related fields.

Verification

To ensure the accuracy of our solutions, it's crucial to verify them. We can achieve this by substituting each solution back into the original equation, ${ x^2 - 7 = 0 }$, and checking if the equation holds true. Let's start with the first solution, ${ x = \sqrt{7} }$. Substituting this value into the equation, we get ${ (\sqrt{7})^2 - 7 = 0 }$. Simplifying the expression, we have ${ 7 - 7 = 0 }$, which is indeed true. This confirms that ${ x = \sqrt{7} }$ is a valid solution. Next, we verify the second solution, ${ x = -\sqrt{7} }$. Substituting this value into the equation, we get ${ (-\sqrt{7})^2 - 7 = 0 }$. When squaring a negative number, the result is positive, so we have ${ 7 - 7 = 0 }$, which is also true. This confirms that ${ x = -\sqrt{7} }$ is also a valid solution. The process of verification is a critical step in problem-solving, especially in mathematics. It ensures that the solutions obtained are correct and that no errors were made during the solution process. By substituting the solutions back into the original equation, we are essentially performing a reverse operation, checking if the steps we took to solve the equation were valid. This process not only confirms the correctness of the solutions but also reinforces the understanding of the underlying mathematical principles. In this case, the verification step highlights the relationship between squaring and taking the square root. It also demonstrates the importance of considering both positive and negative roots when solving equations involving square roots. The fact that both solutions, ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$, satisfy the original equation provides strong evidence that our solution process was accurate and complete. Verification is not just a mechanical step; it's an opportunity to deepen our understanding of the problem and the solution. It allows us to see the connections between different mathematical concepts and to appreciate the elegance and consistency of mathematical reasoning. Moreover, the ability to verify solutions is a valuable skill that can be applied in various contexts, not just in mathematics. It promotes critical thinking and the habit of checking one's work, which are essential for success in any field. The successful verification of our solutions gives us confidence in our results and in our problem-solving abilities.

Conclusion

In conclusion, we have successfully solved the equation ${ x^2 - 7 = 0 }$ and determined the solutions to be ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$. Throughout this process, we have employed fundamental algebraic principles, including isolating the variable term, taking the square root of both sides, and considering both positive and negative roots. The ability to solve such quadratic equations is a crucial skill in mathematics, with applications spanning various fields such as physics, engineering, and computer science. The solutions we obtained represent the values of ${ x }$ that make the equation true, and they also have a geometric interpretation as the points where the graph of the equation intersects the x-axis. Understanding the relationship between algebraic solutions and their graphical representations is essential for developing a comprehensive understanding of mathematics. Furthermore, we have emphasized the importance of verification in problem-solving. By substituting the solutions back into the original equation, we ensured their accuracy and reinforced our understanding of the underlying mathematical concepts. This step is a critical component of the problem-solving process and should not be overlooked. The solutions ${ x = \sqrt{7} }$ and ${ x = -\sqrt{7} }$ are exact solutions, meaning they are represented using the square root symbol, which preserves their precision. While it's possible to approximate these values as decimals, it's generally preferred to leave them in their radical form for greater accuracy. The process of solving this equation has also highlighted the nature of quadratic equations and their solutions. Quadratic equations often have two solutions, which may be real or complex. In this case, we found two distinct real solutions, indicating that the graph of the equation intersects the x-axis at two points. This understanding is valuable for solving other quadratic equations and for analyzing their properties. The skills and concepts we have discussed in this context are applicable to a wide range of mathematical problems. The ability to manipulate equations, solve for unknown variables, and interpret solutions is essential for success in mathematics and related disciplines. Moreover, the emphasis on verification and understanding the underlying principles promotes critical thinking and problem-solving abilities, which are valuable assets in any field of study or profession. Therefore, mastering these concepts is not just about solving a specific equation; it's about developing a broader understanding of mathematics and its applications.