Solve 2x² - 5x + 7 = X² - 2: Solution Set Explained
Hey guys! Ever stumbled upon a quadratic equation that looks like a tangled mess? Well, today we're diving deep into one such equation and unraveling its solution set. We're going to break down the problem step-by-step, making sure you not only understand the how but also the why behind each move. So, buckle up and let's get started!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's take a moment to grasp the fundamentals of quadratic equations. Think of them as the rockstars of the polynomial world – they're recognizable by their highest power being a squared term (x²). A standard quadratic equation usually looks something like this: ax² + bx + c = 0, where 'a', 'b', and 'c' are just constant numbers. These constants play a crucial role in determining the shape and position of the parabola that the equation represents when graphed.
Now, you might be wondering, "Why are these equations so important?" Well, quadratic equations pop up in a ton of real-world scenarios. From calculating the trajectory of a ball thrown in the air to designing the curves of a suspension bridge, these equations are the unsung heroes behind many everyday phenomena. Understanding how to solve them is like unlocking a superpower – you gain the ability to model and predict outcomes in a variety of situations.
Solving a quadratic equation essentially means finding the values of 'x' that make the equation true. These values are also known as the roots or solutions of the equation. Graphically, they represent the points where the parabola intersects the x-axis. A quadratic equation can have two real solutions, one real solution (a repeated root), or no real solutions (complex roots). The number and nature of the solutions are determined by the discriminant, which we'll touch upon later.
Setting Up the Equation: 2x² - 5x + 7 = x² - 2
Alright, let's get our hands dirty with the equation at hand: 2x² - 5x + 7 = x² - 2. The first order of business is to get this equation into the standard form (ax² + bx + c = 0). This makes it much easier to apply our solving techniques. To do this, we need to gather all the terms on one side of the equation, leaving zero on the other side. It's like tidying up a messy room – putting everything in its place.
So, how do we do that? We start by subtracting x² from both sides of the equation. This eliminates the x² term on the right side and brings it over to the left side where the other x² term is waiting. We get: 2x² - x² - 5x + 7 = -2. Next, we simplify by combining the x² terms: x² - 5x + 7 = -2. Now, we need to get rid of that pesky -2 on the right side. We do this by adding 2 to both sides of the equation: x² - 5x + 7 + 2 = 0. Finally, we simplify by adding the constant terms: x² - 5x + 9 = 0. Voila! We've successfully transformed our original equation into the standard quadratic form. We can now clearly identify our coefficients: a = 1, b = -5, and c = 9. These values will be our trusty companions as we embark on our solution-finding journey.
Choosing the Right Method: Quadratic Formula
Now that our equation is in tip-top shape, it's time to choose the right tool for the job – a method to solve it. We have a few options in our arsenal: factoring, completing the square, and the quadratic formula. Factoring is a great option if the equation can be easily broken down into two binomials. Completing the square is a more versatile method, but it can be a bit tedious for some equations. And then we have the quadratic formula – the reliable workhorse that can handle any quadratic equation, no matter how complex it looks.
In our case, factoring doesn't seem like a straightforward option, and completing the square might involve some messy fractions. So, we're going to go with the quadratic formula, our trusty friend that always gets the job done. The quadratic formula is a magical formula that gives us the solutions to any quadratic equation in the standard form. It looks like this: x = (-b ± √(b² - 4ac)) / 2a. It might look intimidating at first glance, but trust me, it's much easier to use than it seems.
Before we plug in our values, let's talk about the discriminant. The discriminant is the part of the formula under the square root: b² - 4ac. It's like a secret decoder that tells us about the nature of the solutions. If the discriminant is positive, we have two distinct real solutions. If it's zero, we have one real solution (a repeated root). And if it's negative, we have no real solutions (complex roots). This is because the square root of a negative number is not a real number. In our equation, the discriminant will help us determine whether we're dealing with real or complex solutions.
Applying the Quadratic Formula
Okay, the moment we've all been waiting for – plugging our values into the quadratic formula! Remember, we have a = 1, b = -5, and c = 9. Let's substitute these values into our formula: x = (-(-5) ± √((-5)² - 4 * 1 * 9)) / (2 * 1). Now, it's just a matter of simplifying the expression step-by-step.
First, let's simplify the terms inside the parentheses and under the square root: x = (5 ± √(25 - 36)) / 2. Next, let's calculate the value under the square root: x = (5 ± √(-11)) / 2. Aha! We've stumbled upon a negative number under the square root. This means our discriminant is negative, and we're dealing with complex solutions. Don't worry, this just means our solutions will involve the imaginary unit 'i', where i = √(-1).
Let's rewrite our expression using 'i': x = (5 ± √(11 * -1)) / 2 = (5 ± √11 * √(-1)) / 2 = (5 ± √11 * i) / 2. Now, we can separate the two solutions: x₁ = (5 + √11 * i) / 2 and x₂ = (5 - √11 * i) / 2. These are our complex solutions! They are complex because they have both a real part (5/2) and an imaginary part (±√11 / 2 * i).
Identifying the Solution Set
We've done the hard work, and now it's time to present our findings – the solution set. The solution set is simply a collection of all the values of 'x' that satisfy our equation. In this case, we have two complex solutions: x₁ = (5 + √11 * i) / 2 and x₂ = (5 - √11 * i) / 2.
We can write the solution set in set notation as: {(5 + √11 * i) / 2, (5 - √11 * i) / 2}. This notation clearly indicates that we have two distinct complex solutions. It's important to remember that these solutions are not real numbers, meaning they cannot be plotted on a standard number line. They exist in the complex number plane, which has both a real and an imaginary axis.
So, there you have it! We've successfully navigated the world of quadratic equations, tackled a tricky problem, and uncovered its complex solutions. Remember, the key to mastering these equations is practice and understanding the underlying concepts. Keep exploring, keep questioning, and keep solving!
Conclusion
Solving quadratic equations can seem daunting at first, but with a systematic approach and the right tools, you can conquer any equation that comes your way. We started by understanding the basics of quadratic equations, then transformed our equation into standard form, chose the quadratic formula as our method of attack, applied the formula meticulously, and finally, identified the solution set. Remember, the journey of a thousand miles begins with a single step – and the journey to mastering mathematics begins with understanding the fundamentals. So, keep practicing, and you'll be solving complex equations like a pro in no time! Keep an eye out for more math adventures, guys!
