Solving Radical Equations A Step By Step Guide For √10x+6 = X+3
In this article, we delve into the step-by-step solution of the equation $\sqrt{10x+6} = x+3$. This type of equation, involving a square root, often appears in algebra and calculus problems. Understanding how to solve it effectively is crucial for students and anyone involved in mathematical problem-solving. We will break down each step, providing clear explanations and insights to ensure a comprehensive understanding. We will not only focus on finding the solutions but also on verifying them to ensure they are valid within the original equation's domain. This meticulous approach will help avoid common pitfalls and strengthen your problem-solving skills.
Introduction to Radical Equations
Radical equations, like the one we are addressing, involve variables inside a radical, most commonly a square root. The key to solving these equations lies in isolating the radical and then eliminating it by raising both sides of the equation to the appropriate power. However, this process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, it's essential to check all potential solutions in the original equation. Our main objective here is to solve the radical equation $\sqrt{10x+6} = x+3$, ensuring we identify all valid solutions and discard any extraneous ones. We will walk through each step meticulously, explaining the logic and reasoning behind each action. This approach will not only provide the answer but also equip you with the skills to tackle similar problems confidently.
Step-by-Step Solution
1. Isolate the Radical
The first step in solving the equation $\sqrt{10x+6} = x+3$ is to isolate the radical term. In this case, the square root term is already isolated on the left side of the equation, which simplifies our initial setup. This is a crucial step because it allows us to eliminate the radical in the next stage. When dealing with more complex equations, you might need to perform algebraic manipulations to isolate the radical, such as adding or subtracting terms from both sides. However, for this particular equation, we can proceed directly to the next step. Remember, the goal is to get the radical term by itself on one side of the equation, making it easier to eliminate and solve for the variable. The isolation of the radical is the foundation for the subsequent steps, ensuring a smooth and accurate solution process.
2. Eliminate the Radical
To eliminate the square root in the equation $\sqrt{10x+6} = x+3$, we square both sides of the equation. This is a fundamental algebraic technique for dealing with radicals. Squaring both sides gives us:
This simplifies to:
Now, we need to expand the right side of the equation. Remember that squaring a binomial means multiplying it by itself. Expanding $(x+3)^2$ gives us:
This step is crucial because it transforms the radical equation into a quadratic equation, which we can then solve using standard algebraic methods. The elimination of the radical is a key step in simplifying the equation and making it solvable. It's important to perform this step carefully, ensuring that both sides of the equation are treated equally to maintain the balance and accuracy of the solution.
3. Rearrange into a Quadratic Equation
Now that we have $10x+6 = x^2 + 6x + 9$, our next step is to rearrange the equation into the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. To do this, we need to move all terms to one side of the equation. We can subtract $10x$ and $6$ from both sides:
Simplifying this, we get:
This is now a standard quadratic equation. Rearranging the equation into this form is essential because it allows us to use various methods to solve for $x$, such as factoring, completing the square, or using the quadratic formula. The rearrangement into a quadratic form is a crucial step in solving the equation, as it sets the stage for applying well-established techniques for finding the roots of the equation.
4. Solve the Quadratic Equation
We now have the quadratic equation $x^2 - 4x + 3 = 0$. There are several methods to solve this, but factoring is often the quickest if the quadratic expression can be factored easily. We are looking for two numbers that multiply to 3 and add to -4. These numbers are -1 and -3. Therefore, we can factor the quadratic as follows:
To find the solutions for $x$, we set each factor equal to zero:
Solving these equations gives us:
So, our potential solutions are $x = 1$ and $x = 3$. Factoring is an efficient method for solving quadratic equations when the factors are easily identifiable. However, it's crucial to remember that not all quadratic equations can be factored, and in such cases, other methods like the quadratic formula or completing the square may be necessary. The process of solving the quadratic equation yields potential solutions, but we must proceed to the next step to verify their validity.
5. Check for Extraneous Solutions
Since we squared both sides of the original equation, it's crucial to check our potential solutions, $x = 1$ and $x = 3$, to ensure they are not extraneous. Extraneous solutions are solutions that satisfy the transformed equation but not the original equation. We substitute each value back into the original equation, $\sqrt{10x+6} = x+3$, to verify.
For x = 1:
This solution is valid.
For x = 3:
This solution is also valid.
Both $x = 1$ and $x = 3$ satisfy the original equation. The verification step is essential in solving radical equations to eliminate any extraneous solutions that may arise from the process of squaring both sides. By substituting the potential solutions back into the original equation, we can confirm their validity and ensure the accuracy of our solution.
Final Answer
Therefore, the solutions to the equation $\sqrt{10x+6} = x+3$ are $x = 1$ and $x = 3$. This corresponds to option (B). The comprehensive step-by-step solution provided here not only gives the answer but also demonstrates the process of solving radical equations, including the crucial step of checking for extraneous solutions. Understanding this process is vital for tackling more complex mathematical problems involving radicals. The final answer, backed by a thorough and meticulous solution, provides confidence in the result and reinforces the understanding of the underlying mathematical principles.
