Solution Of The Equation X² - 7x = -5
Introduction: Tackling Quadratic Equations
When faced with an equation like x² - 7x = -5, it's natural to feel a bit puzzled. But fear not! This is a classic quadratic equation, and we have well-established methods to find its solutions. In this comprehensive guide, we'll walk through the process of solving this equation step by step, explaining the underlying concepts and ensuring you understand not just the how, but also the why behind each step. Our goal is to determine which of the provided options – A. (7 ± √69)/2, B. (7 ± √29)/2, C. (-7 ± √29)/2, or D. (-7 ± √69)/2 – correctly represents the solutions to the equation. Understanding quadratic equations is crucial in various fields, including physics, engineering, and even finance, so let's dive in!
Step 1: Setting the Stage – Transforming the Equation
The first crucial step in solving any quadratic equation is to bring it into its standard form. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. This form is essential because it allows us to apply various methods like factoring, completing the square, or the quadratic formula. In our case, the given equation is x² - 7x = -5. To transform it into the standard form, we need to add 5 to both sides of the equation. This gives us: x² - 7x + 5 = 0. Now, we can clearly identify the coefficients: a = 1, b = -7, and c = 5. Having the equation in this form is a fundamental prerequisite for the next steps in finding the solutions. This initial transformation sets the stage for employing powerful techniques that will lead us to the correct answer.
Step 2: The Quadratic Formula – Our Key Tool
The quadratic formula is a cornerstone of algebra, providing a direct method for solving equations in the standard form ax² + bx + c = 0. This formula is especially valuable when factoring is not straightforward or when the equation involves irrational roots. The formula itself is a concise expression: x = (-b ± √(b² - 4ac)) / 2a. It elegantly encapsulates the relationship between the coefficients of the quadratic equation and its solutions. By substituting the values of a, b, and c from our equation into the quadratic formula, we can systematically determine the roots. This method is universally applicable to any quadratic equation, making it an indispensable tool in mathematical problem-solving. In the subsequent steps, we will apply this formula to our specific equation, x² - 7x + 5 = 0, to unravel its solutions. The power of the quadratic formula lies in its ability to handle a wide range of quadratic equations, regardless of the complexity of their coefficients or the nature of their roots.
Step 3: Applying the Formula – Substituting Values
Now that we have the quadratic formula at our disposal and have identified the coefficients of our equation (a = 1, b = -7, and c = 5), the next step is to carefully substitute these values into the formula. This is a critical step, as accuracy in substitution directly impacts the correctness of the final solution. Recall the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. Replacing a, b, and c with their respective values, we get: x = (-(-7) ± √((-7)² - 4 * 1 * 5)) / (2 * 1). Notice how each coefficient is placed within the formula, paying close attention to signs and operations. This meticulous substitution is the bridge between the general formula and the specific solution to our equation. The next phase involves simplifying this expression, which will lead us closer to the solutions that satisfy the equation. Accurate substitution is not just a mechanical step; it reflects a deep understanding of how the formula relates to the equation's components.
Step 4: Simplifying the Expression – Unveiling the Roots
With the values substituted into the quadratic formula, our next task is to simplify the expression and extract the solutions. Starting with x = (-(-7) ± √((-7)² - 4 * 1 * 5)) / (2 * 1), we begin by simplifying the terms inside the square root and the denominator. The expression becomes: x = (7 ± √(49 - 20)) / 2. Next, we simplify the expression under the square root: x = (7 ± √29) / 2. This simplified form represents the two possible solutions for x, one with the addition operation and the other with the subtraction operation. The ± symbol is a concise way of representing these two distinct roots. The simplification process is crucial because it transforms the complex expression into a clear and understandable form, revealing the numerical values of the solutions. We have now successfully navigated the algebraic manipulations, bringing us to a point where we can directly compare our result with the given options. The square root of 29 remains as √29 because 29 is a prime number and cannot be factored further, leaving our solution in its simplest radical form.
Step 5: Identifying the Solution – Matching the Options
Having simplified the expression to x = (7 ± √29) / 2, we are now in a position to identify the correct solution from the given options. Recall the options:
A. (7 ± √69) / 2 B. (7 ± √29) / 2 C. (-7 ± √29) / 2 D. (-7 ± √69) / 2
By direct comparison, it's clear that our simplified solution, x = (7 ± √29) / 2, matches option B perfectly. This step underscores the importance of accurate simplification and attention to detail throughout the solution process. Matching the solution involves not just finding the correct numerical values but also ensuring that the form of the solution aligns with one of the provided choices. The process of elimination can be a valuable strategy here, as we can definitively rule out options that do not match our derived solution. Option B is the only one that mirrors our result, solidifying it as the correct answer. This final comparison step provides a sense of closure, confirming that our algebraic journey has led us to the accurate solution.
Conclusion: The Answer Revealed
In conclusion, by systematically applying the quadratic formula and simplifying the resulting expression, we have successfully determined the solution to the equation x² - 7x = -5. Our step-by-step journey, from transforming the equation into standard form to substituting values and simplifying, has led us to the solution x = (7 ± √29) / 2. This matches option B from the provided choices. Understanding and mastering the quadratic formula is a fundamental skill in algebra, and this exercise demonstrates its power in solving quadratic equations. Remember, the key to success lies in understanding the underlying principles, careful application of the formula, and meticulous simplification. With these skills, you can confidently tackle a wide range of quadratic equations. The beauty of mathematics lies in its logical progression, where each step builds upon the previous one, ultimately leading to a clear and precise solution. This problem serves as a testament to the elegance and effectiveness of algebraic methods in solving real-world mathematical challenges.
