Solving -9t^2 = 75: A Step-by-Step Solution
Hey everyone! Today, we're diving into a math problem that might seem a bit tricky at first, but don't worry, we'll break it down together. We're going to find the solution set for the equation -9t² = 75. This type of problem involves solving for a variable (in this case, 't') when it's squared and has a negative coefficient. So, let's jump right in and see how to tackle this!
Understanding the Problem
Before we start crunching numbers, let's make sure we understand what the question is asking. When we're asked to find the "solution set," it means we need to find all the values of 't' that make the equation true. In other words, what numbers can we plug in for 't' so that when we multiply them by themselves and then by -9, we get 75? This equation involves a squared term (t²), which means we might have more than one solution. Also, the negative sign and the constant on the right side suggest that we might encounter imaginary numbers, but let's not get ahead of ourselves. We'll take it one step at a time. The key here is to isolate the t² term first. This involves undoing the multiplication by -9. Remember, our goal is to get t² all by itself on one side of the equation, so we can then take the square root to solve for t. Keep in mind that when we take the square root of both sides, we need to consider both the positive and negative roots, as both will satisfy the equation when squared. This is a common step where mistakes can happen, so let's pay close attention to it.
Step-by-Step Solution
Okay, let's get to the actual solving part. Here’s how we can find the solution set for -9t² = 75:
1. Isolate the t² Term
Our first goal is to get the t² term by itself on one side of the equation. To do this, we need to get rid of the -9 that's multiplying it. We can do this by dividing both sides of the equation by -9. This is a crucial step because it simplifies the equation and brings us closer to isolating 't'. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain the balance. So, we divide both sides by -9:
-9t² / -9 = 75 / -9
This simplifies to:
t² = -75/9
Now, we have t² isolated on the left side, which is excellent progress. The right side is a negative fraction, which tells us that our solutions for 't' will involve imaginary numbers, as we can't take the square root of a negative number in the realm of real numbers.
2. Simplify the Fraction
Before we take the square root, let's simplify the fraction -75/9. Both 75 and 9 are divisible by 3. Dividing both the numerator and the denominator by 3, we get:
-75 / 3 = -25 9 / 3 = 3
So, our equation now looks like this:
t² = -25/3
Simplifying fractions makes the numbers smaller and easier to work with, reducing the chances of making calculation errors in the subsequent steps. This step also helps in recognizing perfect squares or factors that can be simplified further when taking the square root.
3. Take the Square Root
Now comes the exciting part – taking the square root of both sides! Remember, when we take the square root of a variable squared, we need to consider both the positive and negative roots. This is because both a positive and a negative number, when squared, will result in a positive number. The square root operation is the inverse of squaring, and it helps us to isolate 't'. So, let's apply the square root to both sides:
√(t²) = ±√(-25/3)
This gives us:
t = ±√(-25/3)
The ± sign is super important here because it indicates that there are two possible solutions: one positive and one negative. Don't forget this, or you'll only find half of the answer!
4. Deal with the Negative Sign
We have a negative sign inside the square root, which means we're dealing with an imaginary number. Remember that the square root of -1 is defined as 'i' (the imaginary unit). So, we can rewrite the square root of -25/3 as:
√(-25/3) = √(25/3 * -1) = √(25/3) * √(-1) = √(25/3) * i
This step is crucial for expressing the solution in its proper form, separating the real and imaginary parts. The imaginary unit 'i' allows us to work with square roots of negative numbers, which are essential in many areas of mathematics and physics.
5. Simplify the Square Root
Now, let's simplify the square root of 25/3. We know that the square root of 25 is 5, so we have:
√(25/3) = √25 / √3 = 5 / √3
To rationalize the denominator (get rid of the square root in the bottom), we multiply both the numerator and the denominator by √3:
(5 / √3) * (√3 / √3) = 5√3 / 3
6. Final Solution
Putting it all together, we have:
t = ± (5√3 / 3) * i
So, the solution set for the equation -9t² = 75 is:
{ ± (5i√3) / 3 }
This means there are two solutions: t = (5i√3) / 3 and t = -(5i√3) / 3. These are complex numbers, which makes sense since we were taking the square root of a negative number.
Checking the Answer
It's always a good idea to check our answer to make sure it's correct. We can do this by plugging our solutions back into the original equation and seeing if they satisfy it. Let's take one of the solutions, t = (5i√3) / 3, and plug it into -9t² = 75:
-9 * ((5i√3) / 3)² = 75
Let's break this down:
(5i√3 / 3)² = (25 * i² * 3) / 9
Remember that i² = -1, so:
(25 * -1 * 3) / 9 = -75 / 9
Now, multiply by -9:
-9 * (-75 / 9) = 75
So, the equation holds true for this solution. You can do the same check for the negative solution, t = -(5i√3) / 3, and you'll find that it also satisfies the equation. This gives us confidence that our solution is correct.
Why This Matters
Solving equations like -9t² = 75 might seem like just an abstract math exercise, but it's actually a fundamental skill that’s used in many real-world applications. Understanding how to manipulate equations, isolate variables, and work with square roots and imaginary numbers is crucial in fields like physics, engineering, computer science, and even finance. For example, in physics, you might encounter similar equations when dealing with oscillatory motion or wave phenomena. In electrical engineering, complex numbers (which involve imaginary units) are used extensively to analyze alternating current circuits. So, mastering these mathematical concepts not only helps you ace your exams but also prepares you for a wide range of career paths.
Common Mistakes to Avoid
When solving equations like this, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer. One frequent mistake is forgetting the ± sign when taking the square root. Remember, both the positive and negative roots can be solutions to the equation, so it's crucial to include both. Another error is mishandling the negative sign inside the square root, which leads to incorrect simplification of imaginary numbers. Make sure you correctly identify and extract the imaginary unit 'i' when necessary. Additionally, errors can occur when simplifying fractions or rationalizing denominators, so take your time and double-check your calculations. Finally, always verify your solution by plugging it back into the original equation to catch any mistakes.
Conclusion
So, there you have it! We've successfully navigated the world of imaginary numbers and square roots to find the solution set for -9t² = 75. Remember, the key is to break the problem down into manageable steps, stay organized, and don't forget those ± signs! Keep practicing, and you'll become a pro at solving these types of equations. Solving equations like -9t² = 75 is a fundamental skill in mathematics, crucial for various fields like physics, engineering, and computer science. Mastering these concepts not only aids in academic success but also prepares you for real-world problem-solving. Keep practicing, and you'll become a pro at solving these types of equations. If you have any questions or want to tackle another math problem, just let me know. Happy solving, guys!
