Solving Quadratic Equations: Finding Real & Imaginary Solutions

Hey guys! Let's dive into the fascinating world of quadratic equations and figure out what kind of solutions the equation -90t² + 63 = -91t² has. Understanding the nature of solutions is a fundamental concept in algebra, and it's super useful for a bunch of real-world problems. We'll break down the process step by step, making it easy to follow along. Buckle up, and let's get started!

Understanding Quadratic Equations

First off, what exactly is a quadratic equation? Simply put, it's an equation that can be written in the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The key characteristic of a quadratic equation is the x² term – it's what makes it quadratic. The solutions to a quadratic equation are the values of 'x' (or in our case, 't') that make the equation true. These solutions are also known as roots or zeros of the equation. These roots can be real numbers (like 1, -2.5, or √3) or complex numbers (which involve the imaginary unit 'i', where i = √-1). The number of solutions and their nature (real or complex) depend on the values of 'a', 'b', and 'c', and they can be found by using a couple of methods that we are going to explore. So, keep reading to master them!

In our given equation, -90t² + 63 = -91t², we can see the presence of the t² term, which tells us it's a quadratic equation, or it can be easily rearranged to fit the form. Because our goal is to find the type of solution for the equation, it is useful to learn some core concepts regarding the topic. The first one we need to know is the discriminant. The discriminant is a part of the quadratic formula, and it tells us a lot about the roots of a quadratic equation without actually solving it. The discriminant is calculated as b² - 4ac. The value of the discriminant determines the nature of the roots. If the discriminant is positive (b² - 4ac > 0), the equation has two distinct real roots. If the discriminant is zero (b² - 4ac = 0), the equation has one real root (a repeated root). If the discriminant is negative (b² - 4ac < 0), the equation has two complex (imaginary) roots. We will use it later in this article. Now, let’s go ahead and rearrange our equation into the standard form to make the solving process easier.

Solving the Equation: Step-by-Step

Now, let's get down to business and solve the equation -90t² + 63 = -91t². Our goal is to find the values of 't' that satisfy this equation. The first step is to rearrange the equation into the standard quadratic form, ax² + bx + c = 0. To do this, let's move all the terms to one side of the equation. Add 91t² to both sides to get -90t² + 91t² + 63 = 0. Combining like terms, we have t² + 63 = 0. Now, we can see that our equation is in the standard form with a = 1, b = 0, and c = 63. Remember the discriminant we talked about earlier? We can apply it here! Let's calculate the discriminant: b² - 4ac = 0² - 4 * 1 * 63 = -252. Since the discriminant is negative (-252), we know that the equation has two complex (imaginary) solutions. We can also directly solve the equation for 't'. Subtract 63 from both sides: t² = -63. Then, take the square root of both sides: t = ±√-63. Since we have a negative number under the square root, we know that the solutions will involve the imaginary unit 'i'. We can simplify this further: t = ±√(63) * √(-1) = ±√(9 * 7) * i = ±3√7i. Therefore, the solutions are t = 3√7i and t = -3√7i. These are two imaginary solutions, which confirms our previous conclusion using the discriminant. Isn't this fantastic? Let's recap the methods and the important concepts for a better grasp.

Analyzing the Solutions

Now that we've solved the equation, let's analyze the solutions we found. We found that the equation -90t² + 63 = -91t² has two imaginary solutions. This means the graph of the equation (if we were to plot it) doesn't intersect the x-axis. Real solutions represent the points where the graph of the equation crosses the x-axis, the points where y=0. However, in this case, since the solutions are imaginary, there are no real x-intercepts. Imaginary solutions always come in conjugate pairs (a + bi and a - bi), where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. In our case, the conjugate pairs are 3√7i and -3√7i. Knowing the nature of the solutions helps us understand the behavior of the quadratic function and is super important in various applications, like physics, engineering, and economics. For instance, in physics, quadratic equations are used to model projectile motion. Depending on the initial conditions (like the initial velocity and angle), the solutions to the quadratic equation may be real (the projectile hits the ground) or imaginary (the projectile doesn't hit the ground). So, understanding the type of solution helps determine the projectile's trajectory. You can see how this math stuff connects to real-world scenarios? It's pretty cool, right? So, let's review the methods and the main concepts. It will help you to retain all the information!

Methods and Concepts Recap

Alright, let's quickly recap what we've covered, guys. We started with the quadratic equation -90t² + 63 = -91t² and wanted to determine the type of solutions it has. First, we rearranged the equation into the standard quadratic form, t² + 63 = 0. Then, we can calculate the discriminant (b² - 4ac) to determine the nature of the roots without actually solving the equation. The discriminant is b² - 4ac. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root (a repeated root). If the discriminant is negative, the equation has two complex (imaginary) roots. We calculated the discriminant, and we found it to be negative. We knew that the equation would have two imaginary solutions. So, the discriminant is a powerful tool to quickly analyze the solutions. However, we went the extra mile and solved the equation for 't' directly. We found that the solutions were t = 3√7i and t = -3√7i, confirming our conclusion. These are two imaginary solutions. Understanding the different types of solutions – real and imaginary – is super important. It helps us interpret the behavior of quadratic equations in various applications. Remember that imaginary solutions occur when the discriminant is negative. Keep in mind that we rearranged the equation into the standard form ax² + bx + c = 0. Also, the discriminant is a part of the quadratic formula, and it tells us a lot about the roots of a quadratic equation without actually solving it. The discriminant is calculated as b² - 4ac. The value of the discriminant determines the nature of the roots. So, guys, you have everything in your toolbox to understand this type of problem. Now you can solve it by yourself!

Conclusion

So, there you have it! The equation -90t² + 63 = -91t² has two imaginary solutions. We arrived at this conclusion by rearranging the equation, calculating the discriminant, and solving for 't' directly. This process highlights the importance of understanding quadratic equations and their solutions. Keep practicing, and you'll become a pro in no time! Remember that you can always go back to this article to review all the important concepts. Keep exploring the world of math, and have fun doing it! Thanks for reading, and see you next time, guys!