Solving The Equation Sqrt(x) + 6 = X A Step-by-Step Guide

Hey guys! Let's dive into this intriguing math problem together. We've got the equation $\sqrt{x}+6=x$, and we need to figure out which of the given options – 4, 9, both 4 and 9, or no solution – is the correct answer. This type of equation, involving a square root, often requires a bit of algebraic maneuvering to solve. The key here is to isolate the square root term and then square both sides of the equation. However, we need to be cautious because squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. So, let's get started and carefully walk through the steps, making sure we verify our answers in the end. Math can be a bit like a puzzle, and it's super satisfying when you finally fit all the pieces together!

Isolating the Square Root

Our first crucial step in solving the equation $\sqrt{x}+6=x$ is to isolate the square root term. Think of it like trying to get the main ingredient ready before you start cooking – you want to make sure it's the focus. To do this, we need to get the $\sqrt{x}$ term all by itself on one side of the equation. The easiest way to achieve this is by subtracting 6 from both sides of the equation. This maintains the balance of the equation, ensuring that we're performing a valid algebraic operation. When we subtract 6 from both sides, the equation transforms from $\sqrt{x}+6=x$ to $\sqrt{x} = x - 6$. Now, we have the square root term nicely isolated, setting us up for the next step in our solution process. Isolating the square root is a fundamental technique in solving equations of this type, and mastering this step will make tackling more complex problems much easier. It's like setting the foundation for a building – a solid first step is essential for success. So, now that we've isolated the square root, we're ready to move on to the next stage of our mathematical journey.

Squaring Both Sides

Now that we've successfully isolated the square root term in our equation, $\sqrt{x} = x - 6$, the next logical step is to eliminate the square root. The most effective way to do this is by squaring both sides of the equation. Remember, whatever operation we perform on one side of the equation, we must also perform on the other side to maintain the balance and ensure the equality remains valid. When we square the left side, $(\sqrt{x})^2$, the square root simply disappears, leaving us with just $x$. However, squaring the right side, $(x - 6)^2$, requires a bit more attention. We need to remember that squaring a binomial means multiplying it by itself: $(x - 6)^2 = (x - 6)(x - 6)$. To expand this, we can use the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last). Multiplying the terms, we get: $(x - 6)(x - 6) = x^2 - 6x - 6x + 36 = x^2 - 12x + 36$. So, squaring both sides of our equation gives us: $x = x^2 - 12x + 36$. This transforms our original equation into a quadratic equation, which we can then solve using various methods. Squaring both sides is a powerful technique, but it's crucial to remember that it can sometimes introduce extraneous solutions, so we'll need to check our answers later. For now, we've successfully eliminated the square root and have a quadratic equation to work with.

Solving the Quadratic Equation

After squaring both sides, we arrived at the quadratic equation $x = x^2 - 12x + 36$. To solve this, the first thing we need to do is set the equation equal to zero. This is a standard procedure for solving quadratic equations, as it allows us to use techniques like factoring or the quadratic formula. To get the equation in the standard form ($ax^2 + bx + c = 0$), we'll subtract $x$ from both sides. This gives us: $0 = x^2 - 12x + 36 - x$, which simplifies to $0 = x^2 - 13x + 36$. Now that we have our quadratic equation in the standard form, we can choose a method to solve it. One common method is factoring. We need to find two numbers that multiply to 36 and add up to -13. After some thought, we can see that -4 and -9 fit the bill: $(-4) * (-9) = 36$ and $(-4) + (-9) = -13$. So, we can factor the quadratic equation as follows: $0 = (x - 4)(x - 9)$. To find the solutions, we set each factor equal to zero: $x - 4 = 0$ or $x - 9 = 0$. Solving these simple equations gives us two potential solutions: $x = 4$ and $x = 9$. However, we're not quite done yet! Remember that squaring both sides earlier could have introduced extraneous solutions. Therefore, we must check these potential solutions in the original equation to see if they are valid.

Checking for Extraneous Solutions

We've arrived at two potential solutions for our equation, $x = 4$ and $x = 9$. But, as we discussed earlier, squaring both sides of an equation can sometimes lead to extraneous solutions, which are solutions that don't actually satisfy the original equation. Therefore, it's absolutely crucial to check each potential solution in the original equation, $\sqrt{x}+6=x$, to ensure they are valid. Let's start with $x = 4$. Substituting this value into the original equation, we get: $\sqrt{4} + 6 = 4$. Simplifying, we have $2 + 6 = 4$, which gives us $8 = 4$. This is clearly not true, so $x = 4$ is an extraneous solution and must be discarded. Now, let's check $x = 9$. Substituting this value into the original equation, we get: $\sqrt{9} + 6 = 9$. Simplifying, we have $3 + 6 = 9$, which gives us $9 = 9$. This is a true statement, so $x = 9$ is a valid solution. By meticulously checking our potential solutions, we've identified that $x = 4$ is an extraneous solution, and only $x = 9$ satisfies the original equation. This step is a vital part of the problem-solving process, especially when dealing with equations involving square roots or other transformations that can introduce extraneous solutions.

Final Answer

Alright, guys! We've journeyed through the steps of solving the equation $\sqrt{x}+6=x$, and it's time to declare our final answer. We started by isolating the square root, then squared both sides to eliminate it, which led us to a quadratic equation. We solved the quadratic equation and found two potential solutions: $x = 4$ and $x = 9$. However, we wisely remembered the crucial step of checking for extraneous solutions. By substituting each potential solution back into the original equation, we discovered that $x = 4$ was an extraneous solution and didn't hold true. On the other hand, $x = 9$ perfectly satisfied the original equation. Therefore, after careful consideration and verification, we can confidently state that the solution to the equation $\sqrt{x}+6=x$ is $x = 9$. So, the correct answer is B. 9. Remember, in math, it's not just about finding an answer; it's about making sure it's the right answer through careful steps and checks. You nailed it!