Solution To The Equation √(-2x - 5) - 4 = X
This article provides a step-by-step solution to the equation √(-2x - 5) - 4 = x, along with a detailed explanation of the process and verification of the solutions. We will explore the algebraic manipulations required to isolate the variable, address potential extraneous solutions, and ultimately arrive at the correct answer. This comprehensive guide aims to enhance your understanding of solving radical equations and related concepts.
Understanding Radical Equations
When tackling mathematical challenges, it's essential to first understand the underlying concepts. Radical equations, in their essence, are equations where the variable resides under a radical symbol, most commonly a square root. The equation we're diving into, √(-2x - 5) - 4 = x, perfectly exemplifies this, showcasing the variable 'x' nestled within a square root. Solving these types of equations demands a methodical approach, a keen eye for detail, and a solid grasp of algebraic principles. Our primary goal is to isolate the radical term, paving the way for the elimination of the radical itself. This often involves strategic algebraic manipulations, such as squaring both sides of the equation. However, a crucial aspect of solving radical equations is the potential for extraneous solutions – those that emerge during the solving process but do not satisfy the original equation. Therefore, meticulous verification is paramount to ensure the accuracy of our results. This involves substituting the obtained solutions back into the initial equation to confirm their validity. By carefully navigating these steps, we can confidently solve radical equations and arrive at the correct solutions. Therefore, we will delve into the intricacies of solving the given equation, highlighting the importance of each step and the underlying mathematical principles at play.
Step 1: Isolating the Radical
In this initial phase of solving the equation √(-2x - 5) - 4 = x, our primary objective is to isolate the radical term. Isolating the radical is a fundamental step in solving radical equations. It simplifies the process of eliminating the radical later on. By isolating the radical term, we effectively set the stage for subsequent operations that will help us solve for the variable. To achieve this, we need to strategically manipulate the equation to get the radical term, √(-2x - 5), by itself on one side. This involves applying inverse operations to eliminate any terms that are hindering its isolation. The equation currently has a '- 4' term on the same side as the radical. To counteract this, we employ the inverse operation of addition. We add 4 to both sides of the equation. This ensures that the equation remains balanced, adhering to the fundamental principles of algebra. Performing this operation, we effectively eliminate the '- 4' from the left side of the equation, thereby moving us closer to our goal of isolating the radical term. The resulting equation after this step is √(-2x - 5) = x + 4. Now, the radical term is isolated on the left side, and we have a clear path forward to eliminate the radical and solve for 'x'. This step exemplifies the importance of understanding inverse operations in equation solving and their role in simplifying complex expressions. With the radical isolated, we are now well-positioned to proceed to the next phase of the solution process.
Step 2: Squaring Both Sides
With the radical term successfully isolated, the next crucial step in solving the equation √(-2x - 5) = x + 4 is to eliminate the square root. This is achieved by employing a fundamental algebraic technique: squaring both sides of the equation. Squaring both sides is a powerful method used to eliminate square roots from equations. The underlying principle is that squaring a square root effectively cancels out the radical, leaving us with the expression inside the radical. However, it is important to recognize that this operation can sometimes introduce extraneous solutions, which we'll address later. By squaring both sides, we transform the equation into a more manageable form that allows us to solve for the variable. When we square the left side, (√(-2x - 5))^2, the square root is eliminated, leaving us with -2x - 5. On the right side, we square the expression (x + 4), which requires careful expansion. (x + 4)^2 expands to (x + 4)(x + 4), which, when multiplied out, yields x^2 + 8x + 16. So, after squaring both sides, our equation becomes -2x - 5 = x^2 + 8x + 16. This transformation is a significant step forward, as we have eliminated the radical and now have a quadratic equation to solve. This quadratic equation can be solved using various techniques, such as factoring, completing the square, or the quadratic formula. However, before we proceed to solve the quadratic equation, it's crucial to remember the potential for extraneous solutions that squaring both sides can introduce. Therefore, we will need to verify our solutions in the original equation to ensure their validity. With the equation now in quadratic form, we can move on to the next step: solving for 'x'.
Step 3: Solving the Quadratic Equation
Having successfully squared both sides of the original equation, we've arrived at a quadratic equation: -2x - 5 = x^2 + 8x + 16. Now, our focus shifts to solving this quadratic equation to find the potential values of 'x'. Solving quadratic equations is a fundamental skill in algebra, and there are several methods we can employ, including factoring, completing the square, and the quadratic formula. The first step in solving this particular quadratic equation is to rearrange it into standard form, which is ax^2 + bx + c = 0. This involves moving all terms to one side of the equation, leaving zero on the other side. To do this, we add 2x and 5 to both sides of the equation. This yields 0 = x^2 + 10x + 21. Now, the quadratic equation is in standard form, making it easier to solve. We can attempt to solve this equation by factoring. Factoring involves finding two binomials that, when multiplied together, give us the quadratic expression. In this case, we look for two numbers that multiply to 21 and add up to 10. These numbers are 7 and 3. Therefore, we can factor the quadratic expression as (x + 7)(x + 3) = 0. With the equation factored, we can apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This means that either x + 7 = 0 or x + 3 = 0. Solving these two linear equations, we find two potential solutions for 'x': x = -7 and x = -3. However, it's crucial to remember that squaring both sides of an equation can introduce extraneous solutions. Therefore, we must verify these potential solutions in the original equation to ensure they are valid. The next step will be to substitute these values back into the original equation and check if they satisfy it. This verification process will help us identify and discard any extraneous solutions, ensuring we arrive at the correct answer.
Step 4: Verifying the Solutions
Having obtained two potential solutions, x = -7 and x = -3, from solving the quadratic equation, the critical next step is to verify these solutions in the original equation: √(-2x - 5) - 4 = x. Verification is an essential process when solving radical equations, as squaring both sides can sometimes introduce extraneous solutions – values that satisfy the transformed equation but not the original one. To verify our solutions, we will substitute each value of 'x' back into the original equation and check if the equation holds true. Let's start with x = -7. Substituting -7 for 'x' in the original equation, we get √(-2(-7) - 5) - 4 = -7. Simplifying the expression inside the square root, we have √(-2(-7) - 5) = √(14 - 5) = √9 = 3. So, the equation becomes 3 - 4 = -7, which simplifies to -1 = -7. This statement is false, indicating that x = -7 is an extraneous solution and not a valid solution to the original equation. Now, let's verify the second potential solution, x = -3. Substituting -3 for 'x' in the original equation, we get √(-2(-3) - 5) - 4 = -3. Simplifying the expression inside the square root, we have √(-2(-3) - 5) = √(6 - 5) = √1 = 1. So, the equation becomes 1 - 4 = -3, which simplifies to -3 = -3. This statement is true, confirming that x = -3 is a valid solution to the original equation. Through this verification process, we have identified that x = -7 is an extraneous solution and x = -3 is a valid solution. This highlights the importance of verification in solving radical equations to ensure the accuracy of our results. The process of substituting the solutions back into the original equation helps us eliminate any extraneous solutions that may have arisen during the solving process.
Final Answer
After a detailed step-by-step solution and rigorous verification, we have determined that the only valid solution to the equation √(-2x - 5) - 4 = x is x = -3. The potential solution x = -7 was identified as an extraneous solution, which did not satisfy the original equation. Therefore, the correct answer is:
C. -3
This comprehensive guide has walked you through the process of solving a radical equation, emphasizing the importance of isolating the radical, squaring both sides, solving the resulting quadratic equation, and, most importantly, verifying the solutions. By understanding and applying these steps, you can confidently tackle similar equations and enhance your problem-solving skills in mathematics.
