Solving Equations: Finding The Value Of W

Hey guys! Let's dive into a classic algebra problem: solving for a variable! We're going to tackle the equation $w - 3 = \sqrt{21 - 5w}$, where w is a real number. This type of problem often pops up in math, and it's super important to understand the steps involved. It combines linear equations with square roots, so we'll need to use a couple of techniques to get to the solution. The core concept here is to isolate the variable w and find the specific value(s) that make the equation true. We'll walk through the process step-by-step, making sure we cover all the bases, including some important checks to avoid any sneaky errors. Ready to get started? Let’s do it!

Step-by-Step Solution: Unveiling the Value of 'w'

Alright, let’s get down to business and solve the equation $w - 3 = \sqrt{21 - 5w}$. We will systematically walk through each stage to make sure we don't miss any steps. Here’s how we're going to approach this problem:

1. Isolate the Square Root: Our first move is always to deal with the square root. To do this, we'll square both sides of the equation. This will get rid of the radical sign and make the equation easier to handle. Remember, whatever we do to one side, we must do to the other to keep things balanced. So, the equation becomes $(w - 3)^2 = (\sqrt{21 - 5w})^2$.
2. Expand and Simplify: Now, let's expand the left side. $(w - 3)^2$ becomes $w^2 - 6w + 9$. The right side simplifies to $21 - 5w$. Now our equation looks like this: $w^2 - 6w + 9 = 21 - 5w$. We're making progress, guys!
3. Rearrange into a Quadratic Equation: We have a quadratic equation now. To solve it, we need to bring everything to one side so that the other side equals zero. Subtract $21$ and add $5w$ from both sides, so we get $w^2 - 6w + 5w + 9 - 21 = 0$, which simplifies to $w^2 - w - 12 = 0$. This is a standard quadratic equation, perfect for factoring or using the quadratic formula.
4. Solve the Quadratic Equation: Now, we have a few options to solve the quadratic equation $w^2 - w - 12 = 0$. We can either factor it, complete the square, or use the quadratic formula. Factoring is often the quickest method when possible. Let's look for two numbers that multiply to -12 and add up to -1. Those numbers are -4 and 3. So, we can factor the equation into $(w - 4)(w + 3) = 0$.
5. Find Possible Solutions: From the factored equation $(w - 4)(w + 3) = 0$, we can see that either $(w - 4) = 0$ or $(w + 3) = 0$. This gives us two potential solutions: $w = 4$ or $w = -3$. Great, we’ve found two possible values for w! But are they both correct? This is where the next step is crucial.

Verifying the Solutions: The Importance of Checking Your Answers

Okay, guys, here’s a super important point: checking your solutions. When dealing with square root equations, it's absolutely essential to plug your solutions back into the original equation to make sure they're valid. Sometimes, squaring both sides can introduce extraneous solutions – solutions that don't actually work in the original equation. So, let’s test our two potential solutions, $w = 4$ and $w = -3$, to see which ones are the real deal.

Let's start with $w = 4$. Substituting into the original equation $w - 3 = \sqrt{21 - 5w}$, we get $4 - 3 = \sqrt{21 - 5(4)}$. This simplifies to $1 = \sqrt{21 - 20}$, or $1 = \sqrt{1}$. Since $\sqrt{1} = 1$, this solution works! Therefore, $w = 4$ is a valid solution. Awesome!

Now, let's check $w = -3$. Substituting this value into the original equation, we have $-3 - 3 = \sqrt{21 - 5(-3)}$. This simplifies to $-6 = \sqrt{21 + 15}$, or $-6 = \sqrt{36}$. Since $\sqrt{36} = 6$, we end up with $-6 = 6$. This is not true. This means that $w = -3$ is an extraneous solution. It appeared during our calculations, but it doesn't satisfy the original equation.

So, after all the hard work, we discovered that only one of our possible solutions is correct. The verification step is what ensures we end up with the right answer. We always must perform this step whenever we solve a square root equation.

Conclusion: The Final Answer and Key Takeaways

Alright, guys, we did it! We successfully solved for w in the equation $w - 3 = \sqrt{21 - 5w}$. After all the steps, from squaring both sides to checking our work, we've found that the only valid solution is $w = 4$. Remember, that in the original equation, we can only have non-negative results for the square root, which is why we must always verify the solutions to eliminate any extra solutions. The number of solutions can vary depending on the equations and conditions given.

Here's a quick recap of the key takeaways:

• Square Both Sides: Always start by squaring both sides to eliminate the square root.
• Simplify and Rearrange: Expand, simplify, and rearrange the equation to get it into a standard form (like a quadratic equation).
• Solve the Quadratic Equation: Use factoring, completing the square, or the quadratic formula to solve.
• Verify Your Solutions: This is critical! Plug each potential solution back into the original equation to make sure it works.

By following these steps, you'll be well-equipped to solve similar problems. Keep practicing, and you'll get better and better at it. You may have noticed the order of operations, the basic equation to start with is always to isolate the radical term on one side of the equation, the second step is to square both sides. Make sure you use the original equation to verify, as it will tell you if the root found is extraneous. Good luck, and keep up the great work, everyone! Let me know if you have any questions!