Solve 4t² - T + 11 = 2: A Step-by-Step Guide
Hey guys! Today, we're going to dive into solving a quadratic equation. Quadratic equations might seem intimidating at first, but with a systematic approach, they're totally manageable. We'll break down the equation step-by-step, so you can follow along easily. Our mission is to find the values of 't' that satisfy the equation 4t² - t + 11 = 2. Let's get started!
Understanding Quadratic Equations
Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. In simpler terms, it's an equation where the highest power of the variable (in our case, 't') is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. Recognizing this form is crucial because it helps us apply the right methods to solve the equation. In our equation, 4t² - t + 11 = 2, we first need to rearrange it into the standard form before we can proceed. This involves moving the constant term from the right side to the left side of the equation, which we'll do in the next step.
Understanding the anatomy of a quadratic equation also means knowing that it can have up to two solutions, also known as roots or zeros. These solutions are the values of 't' that make the equation true. We can find these solutions using various methods, such as factoring, completing the square, or using the quadratic formula. Each method has its own advantages and is suitable for different types of quadratic equations. For instance, factoring is great for equations that can be easily factored, while the quadratic formula is a more general method that works for any quadratic equation. As we move forward, we'll see how these methods apply to our specific equation.
Moreover, the coefficients 'a', 'b', and 'c' play a significant role in determining the nature of the solutions. The discriminant, which is a part of the quadratic formula (b² - 4ac), tells us whether the equation has two distinct real solutions, one real solution (a repeated root), or two complex solutions. This is a key concept to grasp, as it provides insights into the types of answers we should expect. So, with this basic understanding in mind, let's move on to the next step: rearranging our equation into the standard form. This is a crucial step that sets the stage for the solution process.
Step 1: Rearranging the Equation
Our initial equation is 4t² - t + 11 = 2. To get it into the standard form (ax² + bx + c = 0), we need to move the '2' from the right side to the left side. We can do this by subtracting '2' from both sides of the equation. This ensures that we maintain the balance of the equation, which is a fundamental principle in algebra. So, when we subtract '2' from both sides, we get: 4t² - t + 11 - 2 = 2 - 2. This simplifies to 4t² - t + 9 = 0. Now, our equation is in the standard form, which makes it easier to work with.
Having the equation in standard form is not just about aesthetics; it's about setting up the problem for the next steps in the solution process. By identifying 'a', 'b', and 'c', we can easily plug these values into the quadratic formula or use them in other solution methods. In our case, after rearranging, we can clearly see that a = 4, b = -1, and c = 9. These values are crucial for applying the quadratic formula, which is a reliable method for solving any quadratic equation. The quadratic formula is like a universal key that unlocks the solutions to these types of problems.
Furthermore, understanding the importance of the standard form helps in visualizing the quadratic equation as a parabola on a graph. The solutions to the equation represent the points where the parabola intersects the x-axis. If the equation has no real solutions, it means the parabola does not intersect the x-axis. The standard form also helps in determining the vertex of the parabola, which is the highest or lowest point on the curve. This visual representation can provide additional insights into the nature of the solutions. So, now that we have successfully rearranged the equation and identified the coefficients, we are well-prepared to move on to the next step, which involves choosing the best method for solving the equation. Let's see what that entails!
Step 2: Choosing a Solution Method
Now that our equation is in the standard form 4t² - t + 9 = 0, we need to decide on the best method to solve it. There are several options available, including factoring, completing the square, and using the quadratic formula. Each method has its pros and cons, and the best choice depends on the specific equation we're dealing with. Factoring is often the quickest method, but it only works if the quadratic expression can be easily factored. Completing the square is a more general method, but it can be a bit more involved. The quadratic formula, on the other hand, is a foolproof method that works for any quadratic equation, regardless of whether it can be factored or not.
For our equation, 4t² - t + 9 = 0, factoring might not be the most straightforward approach. The coefficients don't lend themselves to easy factorization, so we might spend a lot of time trying to find the right factors without success. Completing the square is a viable option, but it can involve some tricky manipulations with fractions and square roots. Given these considerations, the quadratic formula seems like the most efficient and reliable method for this particular equation. It's a direct, plug-and-chug approach that will give us the solutions without too much hassle.
The quadratic formula is a powerful tool in our mathematical arsenal. It's derived from the process of completing the square and provides a general solution for any quadratic equation in the standard form. The formula is given by: t = (-b ± √(b² - 4ac)) / (2a). This formula might look a bit intimidating at first, but once you get the hang of it, it's quite straightforward to use. All we need to do is identify the values of 'a', 'b', and 'c' from our equation and plug them into the formula. So, with our decision made, let's move on to the next step: applying the quadratic formula to find the solutions to our equation.
Step 3: Applying the Quadratic Formula
We've decided that the quadratic formula is the best way to solve our equation, 4t² - t + 9 = 0. As a quick reminder, the quadratic formula is t = (-b ± √(b² - 4ac)) / (2a). From our equation, we've identified that a = 4, b = -1, and c = 9. Now, it's time to plug these values into the formula and simplify. This is where careful attention to detail is crucial, as a small mistake in the substitution or simplification can lead to incorrect solutions.
First, let's substitute the values into the formula: t = (-(-1) ± √((-1)² - 4 * 4 * 9)) / (2 * 4). Notice how we've carefully replaced each variable with its corresponding value. The double negative in front of 'b' can be a bit tricky, so it's important to pay close attention to signs. Now, let's simplify the expression step-by-step. The numerator becomes (1 ± √(1 - 144)), and the denominator becomes 8. This simplifies further to (1 ± √(-143)) / 8. Uh oh, we've encountered a negative number under the square root! This indicates that our solutions will be complex numbers, which means they will involve the imaginary unit 'i', where i = √(-1).
This is a significant observation. The fact that we have a negative number under the square root tells us that the parabola represented by our quadratic equation does not intersect the x-axis. In other words, there are no real solutions to the equation. Instead, we have two complex solutions. This is a common occurrence in quadratic equations, and it's important to be prepared to handle complex numbers. Now, let's continue simplifying our expression to find the exact complex solutions. We'll rewrite √(-143) as √(143 * -1), which is equal to √143 * √-1, or √143 * i. So, our solutions become t = (1 ± √143 * i) / 8. We've now found the two complex solutions to our quadratic equation!
Step 4: Expressing the Solutions
We've arrived at the solutions t = (1 ± √143 * i) / 8. These are complex numbers, and it's often helpful to express them in the standard form of complex numbers, which is a + bi, where 'a' is the real part and 'b' is the imaginary part. To do this, we simply separate the real and imaginary parts of our solution. So, we can rewrite our solutions as two separate complex numbers:
- t₁ = (1/8) + (√143 / 8)i
- t₂ = (1/8) - (√143 / 8)i
These are the two complex solutions to our quadratic equation 4t² - t + 9 = 0. They are complex conjugates of each other, which is a common occurrence when quadratic equations have complex solutions. The real part of both solutions is 1/8, and the imaginary parts are (√143 / 8) and -(√143 / 8), respectively. Expressing the solutions in this form makes it clear what the real and imaginary components are.
Understanding how to express complex solutions is crucial in mathematics and various fields like engineering and physics. Complex numbers might seem abstract, but they have very real applications in solving problems that involve oscillations, electrical circuits, and quantum mechanics, among others. So, mastering the manipulation of complex numbers is a valuable skill. In our case, we've successfully found the two complex solutions to the quadratic equation and expressed them in the standard form. This concludes our solution process!
Conclusion
Alright guys, we've successfully solved the quadratic equation 4t² - t + 11 = 2! We started by understanding what a quadratic equation is and rearranging it into the standard form. Then, we chose the quadratic formula as our solution method and carefully applied it to find the solutions. Finally, we expressed the complex solutions in the standard form a + bi. This process involved several steps, but by breaking it down, we were able to tackle it effectively. Remember, the key to solving quadratic equations is to follow a systematic approach and pay close attention to detail.
Quadratic equations are a fundamental concept in algebra, and mastering them opens the door to more advanced topics in mathematics. They appear in various contexts, from simple word problems to complex engineering applications. So, the skills we've practiced today will serve us well in the future. Keep practicing, and you'll become a pro at solving quadratic equations in no time! If you have any questions or want to explore more examples, feel free to ask. Happy problem-solving!
