Easy Way To Solve 2x^2 - 11 = 87
Hey guys! Ever come across a math problem that looks a bit intimidating but is actually a piece of cake once you break it down? Today, we're tackling a quadratic equation: 2x² - 11 = 87. Don't let the 'x²' scare you; we'll walk through this step-by-step, and you'll be solving similar problems like a pro in no time. We're aiming to find the values of 'x' that make this equation true. Think of it like finding the secret numbers that unlock the equality. We've got a few options to choose from: A. 7 or -7, B. 2 + 2i or 2 - 2i, and C. 12 or -2. Let's dive in and see which one is the correct solution to our mathematical puzzle. This is all about isolating 'x' and figuring out what it equals. We'll use basic algebraic principles to peel back the layers of the equation and reveal the answers. So, grab your calculators (or just your thinking caps!), and let's get started on demystifying this equation. It's going to be a fun ride, and by the end, you'll have a clearer understanding of how to approach these types of problems. Remember, the key in mathematics is often practice and understanding the fundamental rules. We're going to apply those rules here to find the precise values for 'x'.
Isolating the x² Term
Alright, so our starting point is the equation 2x² - 11 = 87. The first major goal is to get that 'x²' term all by itself on one side of the equation. Think of it as giving 'x²' its own spotlight. To do this, we need to move the '-11' away from it. How do we get rid of a '-11'? You guessed it – we add 11! And remember the golden rule of algebra: whatever you do to one side of the equation, you must do to the other side to keep things balanced. So, we add 11 to both sides:
2x² - 11 + 11 = 87 + 11
This simplifies to:
2x² = 98
See? We're already making progress! The 'x²' term is much closer to being isolated. Now, 'x²' is being multiplied by 2. To undo multiplication, we use division. So, we need to divide both sides of the equation by 2:
(2x²) / 2 = 98 / 2
Which leaves us with:
x² = 49
Boom! We've successfully isolated 'x²'. It's sitting there all by itself, equal to 49. This is a huge step, and it means we're just one step away from finding our values for 'x'. This process of isolating the variable is fundamental in solving many algebraic equations. It involves understanding inverse operations – adding to undo subtraction, subtracting to undo addition, multiplying to undo division, and dividing to undo multiplication. By consistently applying these inverse operations to both sides of the equation, we maintain the equality and systematically simplify the expression until our target variable is alone. It's like a carefully choreographed dance of numbers, where each step brings us closer to the final answer. The beauty of algebra lies in its logic and predictability; when you follow the rules, you're guaranteed to arrive at the correct solution. So, don't shy away from these steps – embrace them as the building blocks of mathematical problem-solving. We've conquered the first half of the battle, and the second half is even more straightforward!
Finding the Value of x
Okay, we're at the point where x² = 49. Now, we need to find out what 'x' actually is. This means we need to perform the inverse operation of squaring. The inverse of squaring a number is taking the square root. So, we need to take the square root of both sides of the equation:
√x² = √49
Now, here's a crucial point that often trips people up: when you take the square root of a number in an equation like this, there are two possible solutions. Why? Because both a positive number and its negative counterpart, when squared, result in a positive number. For example, 7 * 7 = 49, and also (-7) * (-7) = 49. Therefore, 'x' can be either 7 or -7.
So, the solutions are:
x = 7 or x = -7
This means that if you plug either 7 or -7 back into the original equation (2x² - 11 = 87), the equation will hold true. Let's quickly check:
- If x = 7: 2(7)² - 11 = 2(49) - 11 = 98 - 11 = 87. (Correct!)
- If x = -7: 2(-7)² - 11 = 2(49) - 11 = 98 - 11 = 87. (Correct!)
Both values work perfectly. This is why it's essential to remember the '±' (plus or minus) when solving for a variable that has been squared. It's not just about finding one number; it's about finding all numbers that satisfy the condition. This concept extends to many areas of mathematics where roots are involved. Understanding that an equation like x² = k (where k is a positive number) has two real roots, √k and -√k, is fundamental. It ensures we don't miss potential solutions. In more advanced mathematics, you'll encounter complex numbers, which is what option B refers to (involving 'i', the imaginary unit). However, for this particular problem, we are looking for real number solutions, and 7 and -7 are indeed the real numbers that satisfy the equation. So, when you see x², think about both positive and negative possibilities for x. It's a small detail, but it makes a big difference in getting the complete answer. We've officially solved the equation! High fives all around!
Comparing with the Options
So, we've done the math, and we found that the solutions to 2x² - 11 = 87 are x = 7 or x = -7. Now, let's look back at the options provided to see which one matches our findings:
A. 7 or -7 B. 2 + 2i or 2 - 2i C. 12 or -2
Comparing our result, 7 or -7, directly with option A, 7 or -7, it's a perfect match! Option A correctly identifies both the positive and negative square roots of 49. Option B involves imaginary numbers (indicated by 'i'), which arise when solving equations where you need to take the square root of a negative number. Our equation resulted in x² = 49, which has real number solutions, so option B is incorrect. Option C suggests solutions of 12 and -2. Let's quickly check if either of those works, just to be sure. If x = 12, 2(12)² - 11 = 2(144) - 11 = 288 - 11 = 277, which is definitely not 87. If x = -2, 2(-2)² - 11 = 2(4) - 11 = 8 - 11 = -3, also not 87. So, option C is also incorrect.
Therefore, the correct solution to the equation 2x² - 11 = 87 is A. 7 or -7. It's always satisfying when you can match your hard-earned answer to one of the given choices. This confirms that our step-by-step process was accurate and that we didn't miss any crucial details, like remembering both the positive and negative roots. This process of verifying potential solutions is a vital part of the mathematical discipline. It's not enough to just arrive at an answer; you need to be confident that it's the correct answer. By testing our solutions against the original equation and comparing them with the provided options, we've built a strong case for option A being the definitive answer. Keep practicing these problem-solving techniques, and you'll find yourself navigating through algebraic challenges with increasing ease and confidence. It’s all about building that solid foundation, one equation at a time. You guys totally got this!
Final Thoughts
So there you have it, math whizzes! We successfully tackled the quadratic equation 2x² - 11 = 87. We learned that the key is to isolate the 'x²' term first by using inverse operations – adding 11 to both sides to get 2x² = 98, and then dividing by 2 to get x² = 49. The most important part is remembering that when you take the square root to find 'x', you must consider both the positive and negative possibilities. This is because both (7)² and (-7)² equal 49. Thus, the solutions are x = 7 and x = -7. This perfectly matches option A. It’s a great reminder that math often has more than one answer, especially when dealing with squares and roots. Don't forget this little detail next time you see an x²! Practice makes perfect, and the more you work through these equations, the more intuitive they become. Keep challenging yourselves with different problems, and remember to always double-check your work. You've got the tools and the brains to conquer any mathematical puzzle that comes your way. Keep up the awesome work, and happy problem-solving!