Solving Quadratic Equations: Zero Product Property

Hey math enthusiasts! Today, we're diving deep into the world of quadratic equations and, more specifically, how to conquer them using the zero product property. This is a super handy trick, guys, that lets us find the solutions (also known as roots or zeros) of these equations pretty easily. We'll be working through the equation $2x^2 + x - 1 = 2$, breaking it down step by step so you can follow along and nail it. This approach is fundamental, so understanding it will seriously level up your math game!

Understanding the Zero Product Property

Okay, before we jump into the equation, let's get cozy with the zero product property. Simply put, this property states that if the product of two or more factors is zero, then at least one of the factors must be zero. Think of it like this: if you multiply a bunch of numbers and the answer is zero, at least one of those numbers had to be zero. Pretty straightforward, right? This seemingly simple concept is the key to solving a wide range of equations, especially quadratic ones, where we aim to find the values of x that make the equation true. Knowing this property is crucial, and it's the cornerstone of our solution.

Now, how does this help us with an equation like $2x^2 + x - 1 = 2$? Well, the zero product property shines when our equation is set equal to zero. So, our first mission is to transform our given equation into that form. That means we have to make some adjustments to get our equation ready for the magic of the zero product property. We aim to have a quadratic expression equal to zero because that's when we can factor it and apply the property. It's all about setting the stage for easy solving!

This method is super useful because it reduces a complex quadratic equation into simpler linear equations, which we can solve much more easily. It's like breaking down a tough problem into smaller, manageable chunks. The beauty of this method lies in its simplicity. Once you get the hang of it, you can solve these problems pretty quickly. Keep in mind that practice is key, and the more you work through these problems, the more comfortable and confident you'll become. So, get ready to see how it all comes together! The zero product property is your friend, and it will make solving these equations a breeze.

Step-by-Step Solution

Alright, let's get our hands dirty and solve this equation together. Remember our equation: $2x^2 + x - 1 = 2$. Follow these steps to get to the answer.

Step 1: Rearrange the Equation

First things first, we need to get our equation equal to zero. To do that, we'll subtract 2 from both sides. This gives us:

$2x^2 + x - 1 - 2 = 2 - 2$

Which simplifies to:

$2x^2 + x - 3 = 0$

See? Now we have a quadratic equation set equal to zero. This is exactly what we need to use the zero product property later on. This step is crucial because it prepares the equation for factoring, which is our next major step in the process. Without this, we can't apply the zero product property, so it's a non-negotiable step. Think of it as the foundation upon which we build the rest of our solution. We're setting the stage, guys!

Step 2: Factor the Quadratic Expression

Next up, we need to factor the quadratic expression $2x^2 + x - 3$. Factoring means we want to rewrite the expression as a product of two binomials. This might seem a bit tricky at first, but with a little practice, you'll become a pro at it! We are going to factor the quadratic equation. The goal is to break the expression into two parts that, when multiplied together, give us the original expression. This is one of those skills that is super useful in many areas of math, so pay close attention. It is also really important for finding the roots of the equation.

After factoring, we get:

$(2x + 3)(x - 1) = 0$

Awesome, right? We've successfully factored the quadratic expression into two linear factors. If you're not sure how to factor, don't sweat it. There are plenty of resources available to help you, such as online tutorials or your trusty textbook. The important thing is that we've transformed our single quadratic expression into a product of two simpler expressions. This is the heart of the zero product property!

Step 3: Apply the Zero Product Property

Here comes the fun part! Now that we have the product of two factors equal to zero, we can use the zero product property. We know that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x.

First, let's set the first factor equal to zero:

$2x + 3 = 0$

Subtracting 3 from both sides gives us:

$2x = -3$

Dividing both sides by 2, we get:

x = -rac{3}{2}

Now, let's do the same for the second factor:

$x - 1 = 0$

Adding 1 to both sides, we get:

$x = 1$

And there we have it! We've found our solutions. The zero product property really simplifies the whole process. These are the values of x that make the original equation true. Congrats, you've solved your first quadratic equation using the zero product property!

Step 4: State the Solutions

Therefore, the solutions to the equation $2x^2 + x - 1 = 2$ are x = -rac{3}{2} and $x = 1$. The solutions are the values of x that make the equation true. These values satisfy the original equation, meaning that when you substitute them back into the original equation, the equation holds true. Make sure to double-check your work to avoid making careless errors. It's always a good idea to plug your answers back into the original equation to ensure they are correct. Now that you've gone through the process, make sure to practice on more equations. The more you work through these problems, the more comfortable and confident you'll become. So, get ready to see how it all comes together! The zero product property is your friend, and it will make solving these equations a breeze.

Conclusion

So there you have it, guys! We've successfully used the zero product property to find the solutions to the equation $2x^2 + x - 1 = 2$. By rearranging the equation, factoring the quadratic expression, applying the zero product property, and solving for x, we found our solutions. This is a powerful technique that you can apply to a wide range of quadratic equations. Keep practicing, and you'll be solving these equations like a pro in no time! Remember, math is all about practice and understanding the underlying concepts. So keep up the great work, and you'll do great things. Now, go forth and conquer those quadratic equations! You’ve got this! Remember to always double-check your work.

Choosing the Correct Answer

Now, let's look at the multiple-choice options and see which one matches our solutions:

A. $x = -2$ or x = rac{1}{2} B. $x = 1$ or x = rac{3}{2} C. x = -rac{3}{2} or $x = 1$ D. x = -rac{1}{2} or $x = 2$

The correct answer is C. x = -rac{3}{2} or $x = 1$. This is because these are the values we calculated by using the zero product property to solve the equation. Always remember to check your work, but you've done an amazing job if you got to this part. And congrats on mastering another math concept. Make sure to keep practicing. As you grow more confident, you can explore other problem types, like more complex quadratic equations. You'll be surprised at how many problems you can solve with these basic principles.