Solving For W In The Equation 2w^2 + 13w + 34 = (w + 6)^2 A Step-by-Step Guide

Hey guys! Today, we're diving into a fun little math problem where we need to solve for w in the equation 2w^2 + 13w + 34 = (w + 6)^2. Don't worry, it might look a bit intimidating at first, but we'll break it down step by step so it's super easy to follow. We're going to use our algebra skills to simplify the equation, combine like terms, and isolate w. By the end of this, you'll not only know how to solve this specific problem but also have a better handle on solving quadratic equations in general. So, let's grab our pencils and notebooks and get started! Remember, math is like a puzzle, and we're going to find all the pieces and put them together.

Understanding the Equation

Alright, let's take a good look at our equation: 2w^2 + 13w + 34 = (w + 6)^2. The first thing we need to do is understand what we're working with. On the left side, we have a quadratic expression, 2w^2 + 13w + 34. This is a polynomial with a term involving w squared, a term with w, and a constant term. On the right side, we have (w + 6)^2, which is a binomial squared. Our goal is to find the value(s) of w that make this equation true. To do that, we need to simplify both sides and eventually isolate w. This usually involves expanding, combining like terms, and rearranging the equation into a standard form that we can solve. Think of it like balancing a scale; whatever we do to one side, we must do to the other to keep the equation balanced. So, with this in mind, let's roll up our sleeves and start simplifying this equation. We'll begin by expanding the squared term on the right side and then see where that takes us. Remember, the key is to take it one step at a time, and we'll get there together!

Expanding the Right Side of the Equation

Okay, so the next step in solving for w is to expand the right side of our equation, which is (w + 6)^2. Guys, remember that squaring a binomial means multiplying it by itself: (w + 6)^2 = (w + 6)(w + 6). To do this multiplication, we'll use the FOIL method, which stands for First, Outer, Inner, Last. This is a handy way to make sure we multiply each term in the first binomial by each term in the second binomial. So, let's break it down:

• First: Multiply the first terms in each binomial: w * w = w^2
• Outer: Multiply the outer terms: w * 6 = 6w
• Inner: Multiply the inner terms: 6 * w = 6w
• Last: Multiply the last terms: 6 * 6 = 36

Now, let's put it all together: (w + 6)(w + 6) = w^2 + 6w + 6w + 36. We can simplify this further by combining the like terms 6w and 6w, which gives us 12w. So, (w + 6)^2 expands to w^2 + 12w + 36. Now, we've transformed the right side of our equation into a more manageable form. This expansion is a crucial step because it allows us to combine terms with the left side and move closer to isolating w. Next, we'll substitute this expanded form back into our original equation and continue simplifying. We're making good progress, so let's keep the momentum going!

Substituting and Simplifying the Equation

Great job on expanding the right side, guys! Now, let's substitute the expanded form back into our original equation. Remember, our equation was 2w^2 + 13w + 34 = (w + 6)^2, and we found that (w + 6)^2 = w^2 + 12w + 36. So, our equation now becomes 2w^2 + 13w + 34 = w^2 + 12w + 36. The next step is to simplify the equation by moving all the terms to one side. This will help us set the equation equal to zero, which is a standard form for solving quadratic equations. To do this, we'll subtract the terms on the right side from both sides of the equation. Let's start by subtracting w^2 from both sides: 2w^2 - w^2 + 13w + 34 = w^2 - w^2 + 12w + 36, which simplifies to w^2 + 13w + 34 = 12w + 36. Next, let's subtract 12w from both sides: w^2 + 13w - 12w + 34 = 12w - 12w + 36, which simplifies to w^2 + w + 34 = 36. Finally, let's subtract 36 from both sides: w^2 + w + 34 - 36 = 36 - 36, which simplifies to w^2 + w - 2 = 0. Woohoo! We've successfully simplified the equation into a quadratic equation in the standard form ax^2 + bx + c = 0. Now, we're ready to solve for w. We're on the home stretch, guys!

Solving the Quadratic Equation

Awesome work getting the equation into the simplified form w^2 + w - 2 = 0! Now comes the exciting part: solving for w. Since we have a quadratic equation in the form ax^2 + bx + c = 0, we have a few options for solving it. We can try factoring, using the quadratic formula, or completing the square. In this case, factoring looks like the easiest method. Factoring involves finding two binomials that multiply together to give us our quadratic expression. We're looking for two numbers that multiply to -2 (the constant term) and add up to 1 (the coefficient of the w term). Can you think of what those numbers might be? That's right, they are 2 and -1 because 2 * -1 = -2 and 2 + (-1) = 1. So, we can factor the quadratic equation as (w + 2)(w - 1) = 0. Now, here's the key: if the product of two factors is zero, then at least one of the factors must be zero. This is called the zero-product property. So, we set each factor equal to zero and solve for w:

• w + 2 = 0 => w = -2
• w - 1 = 0 => w = 1

And there we have it! We've found two solutions for w: w = -2 and w = 1. These are the values of w that make our original equation true. Give yourselves a pat on the back, guys! We've successfully solved a quadratic equation. Let's move on to verifying our solutions to make sure they are correct.

Verifying the Solutions

Alright, before we declare victory, it's super important to verify our solutions. This step ensures that we didn't make any mistakes along the way and that our values for w actually satisfy the original equation. Remember, we found two possible solutions: w = -2 and w = 1. To verify, we'll plug each value back into the original equation 2w^2 + 13w + 34 = (w + 6)^2 and see if both sides of the equation are equal.

Let's start with w = -2:

• Left side: 2(-2)^2 + 13(-2) + 34 = 2(4) - 26 + 34 = 8 - 26 + 34 = 16
• Right side: (-2 + 6)^2 = (4)^2 = 16

Since both sides are equal (16 = 16), w = -2 is indeed a valid solution.

Now, let's check w = 1:

• Left side: 2(1)^2 + 13(1) + 34 = 2 + 13 + 34 = 49
• Right side: (1 + 6)^2 = (7)^2 = 49

Again, both sides are equal (49 = 49), so w = 1 is also a valid solution.

Woo-hoo! Both of our solutions check out. This gives us confidence that we've correctly solved the equation. Verifying solutions is a crucial step in problem-solving, not just in math but in many areas of life. It's like double-checking your work to ensure accuracy. Great job, guys, for taking the time to verify! We've crossed the finish line with flying colors.

Conclusion

Fantastic work, everyone! We successfully solved for w in the equation 2w^2 + 13w + 34 = (w + 6)^2. We started by expanding the binomial, simplified the equation, factored the quadratic expression, and found two solutions: w = -2 and w = 1. We didn't stop there; we also took the extra step to verify our solutions, confirming that they are indeed correct. Solving this equation involved several important algebraic techniques, including expanding binomials, combining like terms, and factoring quadratic equations. These are skills that will come in handy in many areas of math and science. Remember, the key to solving complex problems is to break them down into smaller, manageable steps. And always, always verify your solutions! I hope you found this walkthrough helpful and that you feel more confident tackling similar problems in the future. Keep practicing, keep exploring, and most importantly, keep having fun with math! You guys are awesome!