Solving The Equation (2x²+x-1)(2x²+x-4)+2=0

Hey everyone! Today, we're going to dive into solving the equation (2x² + x - 1)(2x² + x - 4) + 2 = 0. This one might look a little intimidating at first glance, but trust me, we'll break it down step by step and make it totally manageable. This is a great example of how you can use a bit of clever substitution to simplify a seemingly complex quadratic equation. So, grab your pencils and let's get started! We'll explore different methods to tackle this, making sure you understand the core concepts. The key to solving this type of equation lies in recognizing patterns and making smart substitutions. We will be using algebraic manipulations to turn this equation into something we can easily solve. We will aim to make the process clear, from the initial setup to the final solution. This process will involve a strategic approach that turns a complicated quadratic expression into a much simpler form. By the end of this, you'll be comfortable dealing with similar problems. The approach to solving the equation involves a series of algebraic manipulations. This involves substitution, which simplifies the equation, making it easier to solve. We'll find out the roots of this equation and get the values of x that make it true. This is really exciting stuff, so let's get into it, guys!

Step-by-Step Solution: Unpacking the Equation

Alright, let's get down to business. The equation we're dealing with is (2x² + x - 1)(2x² + x - 4) + 2 = 0. The first thing we want to do is spot the repeating part. Notice that the term (2x² + x) appears in both parts of the equation. This is a HUGE clue! This is where we will start with substitution. This step will make the equation more approachable and simpler to work with. Remember, the goal here is to transform the equation into a more manageable form. So let’s substitute y = 2x² + x. This substitution is the cornerstone of our strategy, simplifying the equation significantly. We can then rewrite our equation as (y - 1)(y - 4) + 2 = 0. Now, this is starting to look much friendlier, right? This looks easier to handle, and you can now see the steps getting more streamlined. The substitution simplifies the structure, enabling straightforward algebraic manipulation. This makes solving the equation less intimidating. We are now closer to finding the solution. This is all about breaking down the complex equation into simpler, manageable parts. The next step is to expand the equation and then further simplify it. This should be an easy and clear process for you all. So keep going, and you'll find the answers you're looking for.

Now, let's expand the expression (y - 1)(y - 4) + 2 = 0. We get: y² - 4y - y + 4 + 2 = 0. This simplifies to y² - 5y + 6 = 0. We've now got a standard quadratic equation in terms of y. This is a much simpler quadratic equation and is easy to solve. So we can factorize it. This makes it easier to find the values of y that satisfy the equation. This step is about simplifying the form, to get to the roots. Now, we are one step closer to solving the equation.

Solving for y: The Quadratic Core

Okay, we've simplified the equation to y² - 5y + 6 = 0. Now, let's solve for y. This is a straightforward quadratic equation that we can solve either by factoring, using the quadratic formula, or completing the square. Factoring is usually the easiest when it works, so let's try that first. What two numbers multiply to 6 and add up to -5? That would be -2 and -3. Therefore, we can factor the equation as (y - 2)(y - 3) = 0. Isn't this neat, guys? We are now close to our answers. With this, we have two possible solutions for y: y = 2 and y = 3. Each of these values will be used to determine the values of x. The goal here is to find the values that make the equation true.

So, we've found our y values. The next part is to find the values for x. Remember that y = 2x² + x. Now we will be working backward. For each value of y, we need to solve for x. This involves substituting each value of y back into our initial substitution equation and solving the resulting quadratic equations for x. We will be going into the final step, and we are almost at the finish line! So let's solve it and get our answers. Now, we're going to put everything back together. Let's do this!

Back-Substitution: Finding the Values of x

Great, now we have the values of y. Let's back-substitute to find the values of x. For y = 2, we have 2x² + x = 2. Rearranging this, we get 2x² + x - 2 = 0. To solve this, we can use the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. In this case, a = 2, b = 1, and c = -2. Plugging these values into the formula, we get: x = (-1 ± √(1² - 42*(-2))) / (22), which simplifies to x = (-1 ± √17) / 4. So for y = 2, we have two solutions for x: x = (-1 + √17) / 4 and x = (-1 - √17) / 4. We are one step away from finishing this problem. It is time to determine the values of x for the second value of y. This involves a similar process to the one we just completed. Once we're done, we will have all of our solutions.

Now, let's solve for y = 3. We have 2x² + x = 3. Rearranging, we get 2x² + x - 3 = 0. Again, we can use the quadratic formula where a = 2, b = 1, and c = -3. Substituting these values, we get: x = (-1 ± √(1² - 42*(-3))) / (22), which simplifies to x = (-1 ± √25) / 4. This simplifies to x = (-1 ± 5) / 4. So we have two more solutions: x = (-1 + 5) / 4 = 1 and x = (-1 - 5) / 4 = -3/2. Fantastic, guys! We have all the solutions for x!

Final Solutions and Conclusion

Alright, let's gather our solutions! We found that the equation (2x² + x - 1)(2x² + x - 4) + 2 = 0 has four solutions. These are: x = (-1 + √17) / 4, x = (-1 - √17) / 4, x = 1, and x = -3/2. Each of these values satisfies the original equation. Each of these values makes the original equation true. We tackled this equation by using substitution to simplify the expression, solve for a new variable, and then back-substitute to find the values of x. This is a common strategy in algebra. The method we used today can be applied to many similar problems. By simplifying the equations, we made it easier to solve them. Great job everyone! You've successfully navigated this quadratic equation. Keep practicing, and you'll become a pro at these problems in no time. If you have any questions or want to try another problem, just let me know! Thanks for joining me today. Keep up the great work, and I will see you soon!