Solving Sqrt(8-4w) = 13 + W A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill, and equations involving radicals often present a unique challenge. This article delves into the process of solving the equation $\sqrt{8-4w} = 13 + w$, providing a step-by-step guide and exploring the underlying concepts. We will not only find the solution(s) to this specific equation but also gain a deeper understanding of how to tackle radical equations in general. The key concepts involved include isolating the radical term, squaring both sides of the equation, solving the resulting quadratic equation, and, crucially, checking for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but not the original radical equation, arising from the process of squaring both sides. Therefore, a thorough verification process is essential to ensure the validity of our solutions.

To begin our journey into solving the equation $\sqrt{8-4w} = 13 + w$, we'll first isolate the radical term. In this particular equation, the radical term, $\sqrt{8-4w}$, is already isolated on the left-hand side, which simplifies our initial steps. Next, we eliminate the square root by squaring both sides of the equation. Squaring both sides is a common technique used to remove radicals from an equation, but it's imperative to remember that this process can introduce extraneous solutions. Therefore, meticulous checking of our solutions against the original equation is an indispensable step in the overall process. Upon squaring both sides, the equation transforms into $8 - 4w = (13 + w)^2$. This transformation is a critical step as it removes the radical, allowing us to work with a more familiar form – a polynomial equation. In the following sections, we will expand and simplify this equation to solve for the unknown variable, $w$. We will then explore methods for solving the resulting quadratic equation, such as factoring, completing the square, or utilizing the quadratic formula. The solutions obtained will then be carefully examined to ascertain their validity within the context of the original radical equation. This process of verification is not merely a formality; it is an integral part of the solution, safeguarding against the inclusion of extraneous solutions.

1. Isolate the Radical

In our equation, $\sqrt{8-4w} = 13 + w$, the radical term $\sqrt{8-4w}$ is already isolated on the left side. This simplifies our first step, as we don't need to perform any algebraic manipulations to isolate the radical. The isolation of the radical is a critical initial step in solving radical equations because it allows us to eliminate the radical by raising both sides of the equation to the appropriate power, which in this case is squaring both sides due to the presence of a square root. By having the radical term isolated, we can proceed to the next step with confidence, knowing that we are working with the equation in its most suitable form for eliminating the radical. This methodical approach is fundamental to effectively solving radical equations and ensuring accurate results. The isolation of the radical term sets the stage for the subsequent steps and highlights the importance of careful observation and algebraic manipulation in solving mathematical equations.

2. Square Both Sides

To eliminate the square root, we square both sides of the equation: $(\sqrt{8-4w})^2 = (13 + w)^2$. Squaring the left side eliminates the square root, resulting in $8 - 4w$. Squaring the right side requires expanding the binomial $(13 + w)^2$, which gives us $169 + 26w + w^2$. Therefore, our equation now becomes $8 - 4w = 169 + 26w + w^2$. This transformation is a pivotal moment in the solution process, as it converts the original radical equation into a quadratic equation, which we can then solve using standard algebraic techniques. However, it is crucial to remember that squaring both sides of an equation can introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. This is because the squaring operation can make a false statement true (e.g., squaring -2 and 2 both give 4, even though -2 ≠ 2). Therefore, after solving for $w$, we must meticulously check our solutions in the original equation to ensure their validity. This step underscores the importance of precision and careful consideration of the potential consequences of algebraic manipulations in solving equations. The process of squaring both sides effectively removes the radical, allowing us to proceed with solving the equation using polynomial techniques, but it simultaneously introduces the need for a thorough verification step to avoid extraneous solutions.

3. Simplify and Rearrange

Now we have the equation $8 - 4w = 169 + 26w + w^2$. To solve for $w$, we need to rearrange this equation into a standard quadratic form, which is $aw^2 + bw + c = 0$. Subtracting $8$ and adding $4w$ to both sides gives us $0 = w^2 + 30w + 161$. This step is crucial because it transforms the equation into a familiar format that we can solve using various methods, such as factoring, completing the square, or the quadratic formula. The process of rearranging terms and setting the equation equal to zero is a fundamental technique in solving polynomial equations, including quadratics. By bringing all terms to one side, we create a clear structure that allows us to identify the coefficients $a$, $b$, and $c$, which are essential for applying the quadratic formula or other solution methods. This systematic approach simplifies the problem and sets the stage for finding the values of $w$ that satisfy the equation. Rearranging the equation into standard form is a critical step in solving quadratic equations, providing a clear path to finding the solution(s).

4. Solve the Quadratic Equation

We now have the quadratic equation $w^2 + 30w + 161 = 0$. To solve this, we can try factoring. We are looking for two numbers that multiply to $161$ and add to $30$. The factors of $161$ are $7$ and $23$, and indeed, $7 + 23 = 30$. Thus, we can factor the quadratic as $(w + 7)(w + 23) = 0$. Setting each factor to zero gives us two potential solutions: $w + 7 = 0$ or $w + 23 = 0$. Solving these linear equations, we find $w = -7$ and $w = -23$. These values are our potential solutions, but remember that we squared both sides of the equation earlier, so we must check for extraneous solutions. Factoring is an efficient method for solving quadratic equations when the quadratic expression can be easily factored. By recognizing the correct factors, we can quickly find the potential solutions. However, the critical next step is to verify these solutions in the original equation to ensure they are not extraneous. This underscores the importance of a thorough verification process when solving equations involving radicals.

5. Check for Extraneous Solutions

Now we must check our potential solutions, $w = -7$ and $w = -23$, in the original equation $\sqrt{8-4w} = 13 + w$.

• For $w = -7$: $\sqrt{8 - 4(-7)} = \sqrt{8 + 28} = \sqrt{36} = 6$. On the other side, $13 + (-7) = 6$. Since $6 = 6$, $w = -7$ is a valid solution.
• For $w = -23$: $\sqrt{8 - 4(-23)} = \sqrt{8 + 92} = \sqrt{100} = 10$. On the other side, $13 + (-23) = -10$. Since $10 \neq -10$, $w = -23$ is an extraneous solution.

The process of checking for extraneous solutions is paramount in solving radical equations. Squaring both sides of an equation can introduce solutions that do not satisfy the original equation, and these extraneous solutions must be identified and discarded. By substituting our potential solutions back into the original equation, we can verify their validity. In this case, $w = -7$ satisfies the equation, while $w = -23$ does not, highlighting the crucial role of the verification step in obtaining the correct solution. This step reinforces the importance of a meticulous and comprehensive approach to solving mathematical equations. The identification and rejection of extraneous solutions are critical for ensuring the accuracy and validity of our final answer.

6. State the Solution

The only valid solution is $w = -7$. The value $w = -23$ is an extraneous solution and is not part of the solution set. Therefore, we have successfully solved the equation $\sqrt{8-4w} = 13 + w$ and found the solution. Stating the solution clearly and explicitly is essential for completing the problem-solving process. It provides a concise answer to the original question and ensures that the result is easily understood. In this case, by clearly stating that $w = -7$ is the only valid solution, we leave no ambiguity about the final answer. This final step of stating the solution reinforces the importance of clear communication and precision in mathematics. A well-stated solution is the culmination of the entire problem-solving process, showcasing the accuracy and completeness of the solution.

In summary, solving the equation $\sqrt{8-4w} = 13 + w$ involves isolating the radical, squaring both sides, simplifying to a quadratic equation, solving for the potential solutions, and, most importantly, checking for extraneous solutions. The only valid solution is $w = -7$. This process demonstrates the importance of careful algebraic manipulation and the necessity of verifying solutions in the original equation when dealing with radical equations. The ability to solve radical equations is a valuable skill in mathematics, with applications in various fields such as physics, engineering, and computer science. By mastering the techniques outlined in this article, you can confidently tackle a wide range of radical equations and enhance your problem-solving abilities. The critical steps of isolating the radical, squaring both sides, simplifying, and checking for extraneous solutions form a systematic approach that ensures accurate and valid solutions. This methodical process not only solves the specific equation at hand but also builds a strong foundation for tackling more complex mathematical problems in the future. The experience gained in solving radical equations contributes to a deeper understanding of algebraic principles and enhances the overall mathematical proficiency of the solver.

Solving radical equations, extraneous solutions, quadratic equation, algebraic manipulation, isolating the radical, squaring both sides, checking solutions, mathematical equations, problem-solving, algebra.