Solving The Equation X = √(x+1) + 5: A Step-by-Step Guide
This article delves into the intricacies of solving equations containing square roots, with a specific focus on the equation x = √(x+1) + 5. We will explore the step-by-step process, underlying principles, and potential pitfalls involved in finding the solution to this type of equation. Understanding how to solve equations with square roots is a fundamental skill in algebra and is essential for various mathematical and scientific applications. Mastering these techniques will empower you to tackle more complex problems and develop a deeper understanding of algebraic manipulations.
The core concept behind solving equations with square roots lies in isolating the radical term and then eliminating it by squaring both sides of the equation. However, this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it is crucial to verify each solution obtained by substituting it back into the original equation. In this article, we will meticulously demonstrate this process, highlighting the importance of verification and providing clear explanations for each step. Let's embark on this journey to unravel the solution to the equation x = √(x+1) + 5 and gain a comprehensive understanding of solving equations with square roots.
1. Isolating the Square Root
The first crucial step in solving the equation x = √(x+1) + 5 is to isolate the square root term. This means we need to get the term containing the square root, which is √(x+1), alone on one side of the equation. To achieve this, we subtract 5 from both sides of the equation. This operation maintains the equality and allows us to manipulate the equation effectively. Subtracting 5 from both sides gives us:
x - 5 = √(x+1) + 5 - 5
Simplifying the equation, we get:
x - 5 = √(x+1)
Now, the square root term, √(x+1), is isolated on the right side of the equation. This isolation is a critical step because it allows us to eliminate the square root by squaring both sides in the next step. Before proceeding, it's important to emphasize the significance of this isolation. Without isolating the square root, squaring both sides would lead to a more complex equation that is difficult to solve. By isolating the square root, we set the stage for a straightforward process of eliminating the radical and solving for x. This step demonstrates a fundamental principle in solving equations with radicals: isolate the radical before eliminating it. This principle ensures a smoother and more manageable solution process.
2. Squaring Both Sides
With the square root term isolated, the next step is to eliminate the square root by squaring both sides of the equation. This is a valid algebraic operation as long as we perform it on both sides, maintaining the equality. Squaring both sides of the equation x - 5 = √(x+1) gives us:
(x - 5)² = (√(x+1))²
On the left side, we have (x - 5)², which we need to expand using the formula (a - b)² = a² - 2ab + b². Applying this formula, we get:
(x - 5)² = x² - 2(x)(5) + 5² = x² - 10x + 25
On the right side, squaring the square root √(x+1) simply eliminates the radical, leaving us with:
(√(x+1))² = x + 1
Therefore, our equation now becomes:
x² - 10x + 25 = x + 1
This equation is a quadratic equation, which we can solve using various methods, such as factoring, completing the square, or the quadratic formula. The act of squaring both sides is a powerful technique for eliminating square roots, but it is crucial to remember that this operation can sometimes introduce extraneous solutions. Extraneous solutions are solutions that satisfy the squared equation but do not satisfy the original equation. Therefore, we must verify our solutions in the original equation to ensure they are valid. This verification step is a critical safeguard against extraneous solutions and ensures the accuracy of our final answer.
3. Simplifying and Rearranging
Now that we have eliminated the square root, we need to simplify and rearrange the equation into a standard form that we can easily solve. Our equation is currently x² - 10x + 25 = x + 1. To solve this, we need to set the equation equal to zero, which is the standard form for a quadratic equation. To do this, we subtract x and 1 from both sides of the equation:
x² - 10x + 25 - x - 1 = x + 1 - x - 1
Simplifying, we get:
x² - 11x + 24 = 0
This is a quadratic equation in the standard form ax² + bx + c = 0, where a = 1, b = -11, and c = 24. Now that we have the equation in this form, we can proceed to solve it. The next step typically involves factoring the quadratic expression, if possible, or using the quadratic formula if factoring is not straightforward. Rearranging the equation into standard form is a crucial step because it allows us to apply well-established methods for solving quadratic equations. This step demonstrates the importance of algebraic manipulation in simplifying equations and preparing them for solution.
4. Solving the Quadratic Equation
We now have the quadratic equation x² - 11x + 24 = 0. To solve this equation, we can attempt to factor it. Factoring involves finding two binomials that multiply together to give the quadratic expression. We are looking for two numbers that multiply to 24 and add up to -11. These numbers are -3 and -8. Therefore, we can factor the quadratic as follows:
(x - 3)(x - 8) = 0
Now, according to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible equations:
x - 3 = 0 or x - 8 = 0
Solving the first equation, x - 3 = 0, we add 3 to both sides:
x = 3
Solving the second equation, x - 8 = 0, we add 8 to both sides:
x = 8
So, we have obtained two potential solutions: x = 3 and x = 8. However, as we discussed earlier, it is crucial to verify these solutions in the original equation because squaring both sides can introduce extraneous solutions. We must substitute each potential solution back into the original equation to check if it satisfies the equation. This verification step is the final and most important step in ensuring we have the correct solution(s).
If factoring the quadratic equation is not straightforward, we can also use the quadratic formula, which is a general formula for finding the roots of any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is:
x = (-b ± √(b² - 4ac)) / 2a
In our case, a = 1, b = -11, and c = 24. Substituting these values into the quadratic formula, we would arrive at the same solutions, x = 3 and x = 8. However, regardless of the method used to solve the quadratic equation, the verification step remains essential.
5. Verifying the Solutions
After solving the quadratic equation, we obtained two potential solutions: x = 3 and x = 8. Now, we must verify these solutions in the original equation, x = √(x+1) + 5, to check for extraneous solutions. This is a critical step because squaring both sides of an equation can sometimes introduce solutions that do not satisfy the original equation.
Let's first verify x = 3. Substitute x = 3 into the original equation:
3 = √(3+1) + 5
3 = √4 + 5
3 = 2 + 5
3 = 7
This statement is false, so x = 3 is not a solution to the original equation. It is an extraneous solution that arose from squaring both sides.
Now, let's verify x = 8. Substitute x = 8 into the original equation:
8 = √(8+1) + 5
8 = √9 + 5
8 = 3 + 5
8 = 8
This statement is true, so x = 8 is a valid solution to the original equation.
Therefore, after verification, we find that only x = 8 is a valid solution, and x = 3 is an extraneous solution. This demonstrates the importance of the verification step in solving equations with square roots. Without verification, we might incorrectly include extraneous solutions in our final answer. The verification process ensures the accuracy and validity of our solutions.
Conclusion
In conclusion, to solve the equation x = √(x+1) + 5, we followed a series of steps: isolating the square root, squaring both sides, simplifying and rearranging the equation into a quadratic form, solving the quadratic equation, and most importantly, verifying the solutions. We found that the potential solutions were x = 3 and x = 8, but after verification, only x = 8 satisfied the original equation. This highlights the crucial role of verification in solving equations with square roots to avoid extraneous solutions.
The process of solving equations with square roots involves several key concepts and techniques. Isolating the radical term is the first critical step, as it allows for the elimination of the square root by squaring both sides. Squaring both sides transforms the equation into a simpler form, often a quadratic equation, which can be solved using various methods such as factoring or the quadratic formula. However, squaring both sides can introduce extraneous solutions, which necessitates the verification step.
The verification step is paramount in ensuring the accuracy of the solution. By substituting the potential solutions back into the original equation, we can identify and discard any extraneous solutions. This step is not merely a formality but an essential part of the problem-solving process.
The equation x = √(x+1) + 5 serves as a valuable example of the intricacies involved in solving equations with square roots. It demonstrates the importance of algebraic manipulation, the potential for extraneous solutions, and the necessity of verification. By mastering these techniques, one can confidently approach and solve a wide range of equations involving square roots. The skills learned in this process are fundamental to algebra and have applications in various fields of mathematics and science.
Therefore, the solution to the equation x = √(x+1) + 5 is:
x = 8
