Solving 5x^2 - 55 = 90 A Step-by-Step Guide
Quadratic equations are fundamental to algebra and appear in various fields, from physics to engineering. Mastering the techniques to solve them is essential for anyone pursuing studies or careers involving mathematics. This comprehensive guide will walk you through the process of solving the quadratic equation 5x² - 55 = 90, providing a step-by-step explanation to ensure a clear understanding of each step involved. By delving into the intricacies of quadratic equations, this article aims to equip you with the knowledge and skills necessary to confidently tackle similar problems.
Understanding Quadratic Equations
Before diving into the solution, let's briefly discuss what quadratic equations are and their general form. A quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable (typically 'x') is 2. The standard form of a quadratic equation is:
ax² + bx + c = 0
Where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. In our case, the equation 5x² - 55 = 90 is a quadratic equation, and we need to manipulate it to find the values of 'x' that satisfy the equation. These values are also known as the roots or solutions of the quadratic equation. Solving quadratic equations involves various techniques, including factoring, completing the square, and using the quadratic formula. Each method has its advantages, and the choice of method often depends on the specific form of the equation. Understanding the nature of quadratic equations and the different methods to solve them is crucial for mastering algebra and its applications.
Step 1: Rearrange the Equation
The first step in solving any quadratic equation is to rearrange it into the standard form ax² + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. Our original equation is:
5x² - 55 = 90
To rearrange it, we need to subtract 90 from both sides of the equation:
5x² - 55 - 90 = 90 - 90
This simplifies to:
5x² - 145 = 0
Now, the equation is in the standard form where a = 5, b = 0 (since there is no 'x' term), and c = -145. Rearranging the equation is a critical first step because it sets the stage for applying various solution methods, such as factoring or using the quadratic formula. Ensuring the equation is in standard form allows for a systematic approach to finding the roots. This step not only prepares the equation for further manipulation but also helps in identifying the coefficients, which are essential for applying methods like the quadratic formula. By transforming the original equation into its standard form, we lay the groundwork for solving it efficiently and accurately.
Step 2: Simplify the Equation
Before proceeding further, it's often helpful to simplify the equation if possible. In our case, we can see that all terms in the equation 5x² - 145 = 0 are divisible by 5. Dividing the entire equation by 5 will simplify the coefficients and make the equation easier to work with. This simplification does not change the solutions of the equation but makes the subsequent steps more manageable. When dealing with quadratic equations, looking for opportunities to simplify can save time and reduce the chances of making errors in calculations. The technique of dividing by a common factor is a fundamental algebraic manipulation that can be applied to various types of equations, not just quadratic ones.
Dividing both sides of the equation by 5, we get:
(5x² / 5) - (145 / 5) = 0 / 5
This simplifies to:
x² - 29 = 0
Now, our equation is much simpler, with a = 1, b = 0, and c = -29. Simplifying the equation not only makes the numbers smaller and easier to handle but also often reveals the underlying structure of the equation more clearly. This can lead to a more intuitive understanding of the problem and potentially suggest the most efficient method for finding the solutions. By simplifying the equation, we pave the way for a more straightforward application of solution techniques, such as taking the square root in this particular case.
Step 3: Isolate the x² Term
Now that we have the simplified equation x² - 29 = 0, the next step is to isolate the x² term. This means we want to get the x² term alone on one side of the equation. To do this, we add 29 to both sides of the equation:
x² - 29 + 29 = 0 + 29
This simplifies to:
x² = 29
Isolating the x² term is a crucial step because it allows us to directly apply the square root property to solve for x. This technique is particularly useful when the quadratic equation is in the form x² = k, where k is a constant. Isolating the variable term is a common strategy in solving various types of equations, as it helps to simplify the problem and make it more amenable to solution. In the context of quadratic equations, isolating the x² term sets up the final step of taking the square root, which will yield the solutions to the equation. By isolating x², we transform the equation into a form that is easily solvable, highlighting the importance of algebraic manipulation in equation solving.
Step 4: Take the Square Root of Both Sides
With the x² term isolated, we can now solve for x by taking the square root of both sides of the equation x² = 29. Remember that when taking the square root, we need to consider both the positive and negative roots because both positive and negative numbers, when squared, will result in a positive value. This is a crucial concept in solving quadratic equations, as it ensures that we find all possible solutions. Failing to consider both roots can lead to an incomplete solution set.
Taking the square root of both sides gives us:
√(x²) = ±√29
This simplifies to:
x = ±√29
The symbol '±' indicates that there are two possible solutions: one positive and one negative. Therefore, the solutions are x = √29 and x = -√29. Considering both the positive and negative roots is a fundamental aspect of solving quadratic equations, as it stems from the definition of the square root operation. The square root of a number is a value that, when multiplied by itself, gives the original number. Since both positive and negative values satisfy this condition, we must include both in our solutions. By taking both the positive and negative square roots, we ensure a complete and accurate solution to the quadratic equation.
Step 5: Approximate the Solutions (Optional)
While x = ±√29 are the exact solutions, it is often useful to approximate these values as decimals, especially in practical applications. The square root of 29 is not a perfect square, so its decimal representation is an irrational number, meaning it goes on infinitely without repeating. Therefore, we typically use a calculator to find an approximate decimal value.
Using a calculator, we find that:
√29 ≈ 5.385
Therefore, the approximate solutions are:
x ≈ 5.385 and x ≈ -5.385
Approximating solutions is particularly useful when the exact solutions are irrational numbers, as it provides a more tangible understanding of the values of x. In real-world scenarios, these decimal approximations can be more meaningful and easier to interpret. While the exact solutions give the precise values that satisfy the equation, the approximate solutions offer a practical way to work with these values in calculations and applications. The process of approximating solutions involves using calculators or numerical methods to find decimal representations of irrational numbers, allowing us to express the solutions in a more usable format. By approximating the solutions, we bridge the gap between the abstract mathematical solutions and their practical implications.
Summary of Steps
To recap, here are the steps we took to solve the quadratic equation 5x² - 55 = 90:
- Rearrange the Equation: Rewrite the equation in the standard form ax² + bx + c = 0.
- Simplify the Equation: Divide the equation by a common factor if possible.
- Isolate the x² Term: Get the x² term alone on one side of the equation.
- Take the Square Root of Both Sides: Remember to consider both positive and negative roots.
- Approximate the Solutions (Optional): Use a calculator to find decimal approximations of the solutions.
These steps provide a systematic approach to solving quadratic equations of this type. The process of solving quadratic equations is a fundamental skill in algebra, and mastering it opens doors to more advanced mathematical concepts and applications. By following these steps, you can confidently tackle a wide range of quadratic equations and understand the underlying principles behind the solutions. The ability to solve quadratic equations is not only essential for academic success but also for practical problem-solving in various fields, making it a valuable skill to acquire. This structured approach ensures clarity and accuracy in finding the solutions, emphasizing the importance of each step in the process.
Conclusion
In this guide, we have thoroughly explained the process of solving the quadratic equation 5x² - 55 = 90. By following the step-by-step instructions, you can confidently solve similar quadratic equations. Remember to rearrange, simplify, isolate the x² term, take the square root of both sides (considering both positive and negative roots), and approximate the solutions if necessary. Quadratic equations are a crucial part of algebra, and mastering them will undoubtedly benefit your mathematical journey. The process outlined in this guide provides a solid foundation for tackling more complex problems and understanding the broader applications of quadratic equations in various fields. By practicing and applying these techniques, you can develop a strong proficiency in solving quadratic equations, which is a valuable skill in both academic and practical settings. This comprehensive approach ensures a deep understanding of the solution process, empowering you to confidently address quadratic equations in different contexts.
