Solving \(\sqrt{3a+1} + \sqrt{a} = 3\) A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving equations involving square roots, with a specific focus on the equation ${\sqrt{3a+1} + \sqrt{a} = 3}$. Equations of this nature often require a systematic approach to isolate the variable and eliminate the radical signs. This guide will walk you through each step, providing explanations and justifications to ensure a clear understanding of the solution process. Understanding how to solve equations such as ${\sqrt{3a+1} + \sqrt{a} = 3}$ is a fundamental skill in algebra and calculus. These types of equations often appear in various mathematical contexts, from basic algebra problems to more complex applications in physics and engineering. The presence of square roots adds a layer of complexity, requiring a strategic approach to isolate the variable and eliminate the radicals. This article aims to provide a comprehensive guide, breaking down each step and explaining the underlying principles. Before we dive into the specifics, it's important to grasp the general strategy for solving radical equations. The primary goal is to eliminate the square roots by squaring both sides of the equation. However, this must be done carefully, as squaring can introduce extraneous solutions. Therefore, it's crucial to verify the solutions obtained at the end of the process. The process typically involves isolating one of the square root terms, squaring both sides, simplifying the resulting equation, and repeating the process if necessary. By meticulously following these steps, we can arrive at the correct solution while avoiding common pitfalls. The equation ${\sqrt{3a+1} + \sqrt{a} = 3}$ serves as an excellent example to illustrate these techniques. It combines two square root terms, making it a bit more challenging than simpler radical equations. By working through this example, you will gain a deeper understanding of the methods involved and develop the confidence to tackle similar problems. This article will not only provide the solution but also emphasize the importance of checking for extraneous solutions, a critical step often overlooked but essential for accuracy. Through detailed explanations and step-by-step guidance, we aim to make the solution process clear and accessible, enhancing your problem-solving skills in algebra. So, let's embark on this journey of solving radical equations and discover the solution to ${\sqrt{3a+1} + \sqrt{a} = 3}$.

Step-by-Step Solution

To begin, our initial equation is ${\sqrt{3a+1} + \/a = 3}$. The first crucial step in solving equations with square roots is to isolate one of the radicals on one side of the equation. This is essential because squaring both sides will eliminate the isolated radical. In our case, we can isolate ${\sqrt{3a+1}}$ by subtracting ${\sqrt{a}}$ from both sides. This gives us the equation ${\sqrt{3a+1} = 3 - \sqrt{a}}$. Isolating the square root is a strategic move that simplifies the equation and sets the stage for the next step. By having a single square root on one side, we can eliminate it by squaring both sides, which is a fundamental technique in solving radical equations. This step is not just about algebraic manipulation; it's about transforming the equation into a more manageable form. Without isolating the radical, squaring both sides would result in a more complex expression due to the cross-terms that arise from squaring a binomial. Isolating the radical allows us to eliminate it directly, paving the way for a simpler equation that can be solved using standard algebraic techniques. In this particular equation, isolating ${\sqrt{3a+1}}$ makes the subsequent steps much clearer and easier to follow. It's a demonstration of how a well-chosen initial step can significantly simplify the entire problem-solving process. The ability to recognize and execute this step effectively is a key skill in mastering radical equations. By isolating the radical, we set the stage for the next critical step: squaring both sides of the equation to eliminate the square root. This will allow us to transform the equation into a more familiar algebraic form, which we can then solve using standard methods. The equation ${\sqrt{3a+1} = 3 - \sqrt{a}}$ is now primed for the next step, which will further simplify our quest to find the value of 'a'.

Next, we square both sides of the equation ${\sqrt{3a+1} = 3 - \sqrt{a}}$. Squaring both sides is a critical step as it eliminates the square root on the left side. This yields ${(\sqrt{3a+1})^2 = (3 - \sqrt{a})^2}$, which simplifies to ${3a+1 = (3 - \sqrt{a})^2}$. Now, we need to expand the right side of the equation. Recall that ${(x - y)^2 = x^2 - 2xy + y^2}$. Applying this formula, we expand ${(3 - \sqrt{a})^2}$ as ${3^2 - 2(3)(\sqrt{a}) + (\sqrt{a})^2}$, which further simplifies to ${9 - 6\sqrt{a} + a}$. Therefore, our equation becomes ${3a+1 = 9 - 6\sqrt{a} + a}$. Squaring both sides is a fundamental technique in solving radical equations, and it's essential to perform this step accurately to avoid errors. The expansion of the squared binomial is a common area where mistakes can occur, so it's crucial to apply the formula correctly. The result of this step is a new equation that, while more complex looking, has eliminated one of the square roots, bringing us closer to isolating the variable 'a'. The presence of the term ${-6\sqrt{a}}$ indicates that we will need to repeat the process of isolating the radical and squaring both sides. However, before we do that, it's advantageous to simplify the equation by combining like terms. This will make the subsequent steps easier to manage. The equation ${3a+1 = 9 - 6\sqrt{a} + a}$ now presents a clear path forward. By rearranging the terms and isolating the remaining square root, we can once again square both sides and eliminate the radical. This iterative process is a hallmark of solving radical equations, requiring careful attention to detail and a methodical approach. The ultimate goal is to transform the equation into a form that can be solved using standard algebraic methods, leading us to the solution for 'a'.

Now, let's simplify the equation ${3a+1 = 9 - 6\sqrt{a} + a}$. To simplify, we want to isolate the remaining square root term. First, subtract 'a' and 1 from both sides of the equation: ${3a + 1 - a - 1 = 9 - 6\sqrt{a} + a - a - 1}$. This simplifies to ${2a = 8 - 6\sqrt{a}}$. Next, we isolate the term with the square root. Subtract 8 from both sides: ${2a - 8 = -6\sqrt{a}}$. Now, divide both sides by -2 to simplify further: ${\frac{2a - 8}{-2} = \frac{-6\sqrt{a}}{-2}}$, which simplifies to ${-a + 4 = 3\sqrt{a}}$. Simplifying the equation is a crucial step in making it more manageable. By isolating the square root term, we set the stage for the next squaring operation, which will eliminate the radical. The process of rearranging terms, combining like terms, and dividing by common factors is a fundamental aspect of algebraic manipulation. Each step in this simplification process brings us closer to a form where we can easily eliminate the square root and solve for the variable 'a'. The simplified equation, ${-a + 4 = 3\sqrt{a}}$, is now in a much cleaner form compared to the original equation. This makes the subsequent steps, such as squaring both sides, less prone to errors. Simplifying equations not only makes them easier to solve but also provides a clearer picture of the relationships between the variables. It's a vital skill in algebra that helps in understanding the structure of equations and choosing the most efficient solution strategies. The equation ${-a + 4 = 3\sqrt{a}}$ now represents a significant milestone in our problem-solving journey. With the square root term isolated, we are well-positioned to apply the squaring operation and move closer to finding the value of 'a'. This iterative process of simplifying and isolating radicals is a common theme in solving radical equations, highlighting the importance of methodical and careful algebraic manipulation.

With the simplified equation ${-a + 4 = 3\sqrt{a}}$, we now square both sides again. Squaring both sides gives us ${(-a + 4)^2 = (3\sqrt{a})^2}$. Expanding the left side using the formula ${(x - y)^2 = x^2 - 2xy + y^2}$, we get ${a^2 - 8a + 16}$. On the right side, we have ${(3\sqrt{a})^2 = 9a}$. Thus, the equation becomes ${a^2 - 8a + 16 = 9a}$. Now, we rearrange the equation to set it to zero, which is a standard form for solving quadratic equations. Subtract 9a from both sides: ${a^2 - 8a + 16 - 9a = 0}$, which simplifies to ${a^2 - 17a + 16 = 0}$. Squaring both sides is a critical step in eliminating the remaining square root. It transforms the equation into a polynomial form, specifically a quadratic equation in this case. The expansion of the binomial on the left side and the simplification on the right side are essential to ensure the accuracy of the resulting equation. The equation ${a^2 - 17a + 16 = 0}$ is a standard quadratic equation, which we can solve using various methods such as factoring, completing the square, or the quadratic formula. The rearrangement of terms to set the equation to zero is a fundamental technique in solving polynomial equations, providing a clear pathway to finding the roots of the equation. The transition from a radical equation to a quadratic equation demonstrates the power of algebraic manipulation in transforming complex problems into more familiar forms. The ability to recognize and apply these transformations is a key skill in problem-solving. The equation ${a^2 - 17a + 16 = 0}$ now represents a significant milestone in our journey to solve the original radical equation. We have successfully eliminated the square roots and obtained a quadratic equation, which we can now solve using standard techniques. This quadratic equation will provide us with potential solutions for 'a', which we will need to verify in the original equation to ensure they are not extraneous solutions. The next step involves solving this quadratic equation, a process that will further reveal the value of 'a'.

We now solve the quadratic equation ${a^2 - 17a + 16 = 0}$. We can solve this by factoring. We look for two numbers that multiply to 16 and add up to -17. These numbers are -1 and -16. So, we can factor the quadratic equation as ${(a - 1)(a - 16) = 0}$. Setting each factor equal to zero gives us the potential solutions ${a = 1}$ and ${a = 16}$. Solving the quadratic equation is a crucial step in finding the potential solutions for the variable 'a'. Factoring is an efficient method when the quadratic equation can be easily factored. The process of identifying the factors involves understanding the relationship between the coefficients of the quadratic equation and the roots. The potential solutions obtained from factoring must be verified in the original equation to ensure they are valid and not extraneous. Extraneous solutions can arise when squaring both sides of an equation, which can introduce roots that do not satisfy the original equation. The solutions ${a = 1}$ and ${a = 16}$ are now candidates for the solution of the original radical equation. However, we must emphasize the importance of verification. This step is often overlooked but is essential to ensure the accuracy of the solution. The process of solving quadratic equations is a fundamental skill in algebra, and the ability to factor quadratic equations efficiently is a valuable tool in problem-solving. The equation ${(a - 1)(a - 16) = 0}$ represents a clear pathway to the potential solutions, but the journey is not complete until we verify these solutions in the original equation. This verification step will confirm whether these values of 'a' are indeed valid solutions or extraneous roots. The next step involves substituting these values back into the original equation to check their validity.

Verification of Solutions

Verification is a critical step to ensure that the solutions we obtained are valid and not extraneous. We have two potential solutions: ${a = 1}$ and ${a = 16}$. Let's start by verifying ${a = 1}$ in the original equation ${\sqrt{3a+1} + \sqrt{a} = 3}$. Substituting ${a = 1}$, we get ${\sqrt{3(1)+1} + \sqrt{1} = 3}$, which simplifies to ${\sqrt{4} + 1 = 3}$, and further to ${2 + 1 = 3}$. This is true, so ${a = 1}$ is a valid solution. Now, let's verify ${a = 16}$. Substituting ${a = 16}$ into the original equation, we get ${\sqrt{3(16)+1} + \sqrt{16} = 3}$, which simplifies to ${\sqrt{49} + 4 = 3}$, and further to ${7 + 4 = 3}$. This simplifies to ${11 = 3}$, which is false. Therefore, ${a = 16}$ is an extraneous solution and not a valid solution to the original equation. Verifying the solutions is a non-negotiable step in solving radical equations. The act of squaring both sides can introduce extraneous solutions, which are values that satisfy the transformed equation but not the original equation. The verification process involves substituting each potential solution back into the original equation to check if it holds true. The substitution and simplification must be performed carefully to avoid errors in determining the validity of the solutions. The solution ${a = 1}$ successfully verified in the original equation, confirming its validity. However, the solution ${a = 16}$ failed the verification, highlighting the importance of this step in discarding extraneous solutions. The extraneous solution arose due to the squaring operation, which can change the domain of the equation and introduce solutions that were not present in the original equation. The verification process provides a critical safeguard against accepting incorrect solutions and ensures the accuracy of the final answer. The equation ${\sqrt{3a+1} + \sqrt{a} = 3}$ has been thoroughly analyzed, and the verification step has played a pivotal role in identifying the true solution. The final result of the verification process is that ${a = 1}$ is the only valid solution, and ${a = 16}$ is an extraneous solution. This underscores the importance of a meticulous approach in solving radical equations, where each step must be carefully executed and the solutions rigorously verified. The next step involves stating the final solution based on the verification process.

Final Answer

After verifying the potential solutions, we found that ${a = 1}$ is the only valid solution, while ${a = 16}$ is an extraneous solution. Therefore, the solution to the equation ${\sqrt{3a+1} + \sqrt{a} = 3}$ is ${a = 1}$. Stating the final answer is the culmination of the entire problem-solving process. It involves clearly presenting the solution that has been verified and confirmed to be correct. The final answer should be concise and unambiguous, leaving no doubt about the solution to the original equation. The solution ${a = 1}$ represents the value that satisfies the original radical equation, as confirmed through the verification process. The extraneous solution, ${a = 16}$, was discarded because it did not satisfy the original equation, underscoring the importance of the verification step. The equation ${\sqrt{3a+1} + \sqrt{a} = 3}$ has been successfully solved, and the final answer, ${a = 1}$, represents the value of 'a' that makes the equation true. This comprehensive solution process demonstrates the steps involved in solving radical equations, including isolating the radical, squaring both sides, simplifying the equation, solving the resulting equation, and verifying the solutions. The final answer is a testament to the methodical approach and the careful execution of each step, resulting in an accurate and verified solution. This journey through solving the radical equation ${\sqrt{3a+1} + \sqrt{a} = 3}$ provides valuable insights into the techniques and considerations necessary for solving similar problems. The final answer, ${a = 1}$, stands as the correct solution, validated by a rigorous verification process.

Conclusion

In conclusion, solving the equation ${\sqrt{3a+1} + \sqrt{a} = 3}$ involves a multi-step process that includes isolating the square root, squaring both sides, simplifying the equation, solving the resulting quadratic equation, and crucially, verifying the solutions. The extraneous solution ${a = 16}$ highlights the importance of the verification step, and the final valid solution is ${a = 1}$. Understanding the methodologies involved in solving equations with square roots is vital for success in algebra and related fields. The process demonstrated in this article provides a clear and systematic approach to tackling such problems. The strategic steps of isolating radicals, squaring both sides, and simplifying the equation are fundamental techniques in algebraic manipulation. The resulting quadratic equation requires skills in factoring or other methods of solving quadratic equations, further emphasizing the interconnectedness of algebraic concepts. The verification step is the cornerstone of the solution process for radical equations. It serves as a crucial check against extraneous solutions that can arise from the squaring operation. The extraneous solution ${a = 16}$ in this example vividly illustrates why verification is essential. The final valid solution, ${a = 1}$, represents the accurate answer that satisfies the original equation. This solution is the culmination of a careful and methodical approach, where each step has been executed with precision and diligence. The ability to solve equations like ${\sqrt{3a+1} + \sqrt{a} = 3}$ equips students and practitioners with a powerful tool for problem-solving in various contexts. Radical equations appear in numerous mathematical applications, and mastering their solution is a valuable skill. The comprehensive guide provided in this article aims to empower readers with the knowledge and confidence to tackle similar challenges. From the initial isolation of radicals to the final verification of solutions, each step is designed to ensure accuracy and understanding. The experience of solving this equation not only yields a specific solution but also reinforces the broader principles of algebraic problem-solving. The equation ${\sqrt{3a+1} + \sqrt{a} = 3}$ serves as an excellent example to illustrate these principles, providing a practical foundation for more advanced mathematical explorations. The final solution, ${a = 1}$, is a testament to the effectiveness of the techniques and strategies discussed in this article. This conclusion reiterates the importance of a systematic approach, the necessity of verification, and the value of mastering algebraic skills for solving radical equations.