Solving -x = -7/2 & Simplifying √{89 - 28√10}

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Hey guys! Let's dive into some math problems today. We're going to tackle solving the equation -x = -7/2 and simplifying the expression √{89 - 28√10}. Both of these problems involve different areas of math, but they're totally manageable if we break them down step by step. So, grab your pencils, and let's get started!

Solving -x = -7/2

When we look at the equation -x = -7/2, our main goal is to isolate x on one side of the equation. This means we need to get rid of that negative sign in front of the x. How do we do that? Well, we can multiply both sides of the equation by -1. This is a fundamental algebraic principle: whatever you do to one side of the equation, you must do to the other to keep it balanced.

Let's walk through it. We start with -x = -7/2. Multiply both sides by -1:

(-1) * (-x) = (-1) * (-7/2)

On the left side, a negative times a negative is a positive, so (-1) * (-x) becomes x. On the right side, we also have a negative times a negative, so (-1) * (-7/2) becomes 7/2. Therefore, our equation simplifies to:

x = 7/2

Now, we have solved for x! x is equal to 7/2. If we want to express this as a mixed number, we can divide 7 by 2. 2 goes into 7 three times (2 * 3 = 6), with a remainder of 1. So, 7/2 is the same as 3 and 1/2, or 3.5 in decimal form. Therefore, the solution to the equation -x = -7/2 is x = 7/2, which is also equal to 3.5. It's pretty straightforward once you understand the basic algebraic manipulation.

Understanding the Basics of Algebraic Equations

Solving equations like -x = -7/2 relies on understanding a few core principles of algebra. The most important concept here is maintaining balance. Imagine an equation as a balanced scale. Whatever you do to one side, you must do to the other to keep the scale balanced. This principle allows us to manipulate equations to isolate variables and find solutions.

Another key concept is the idea of inverse operations. To isolate a variable, we use operations that "undo" the operations currently acting on it. In our example, the variable x is being multiplied by -1. The inverse operation of multiplication is division, but in this case, multiplying by -1 again is the most direct route. By understanding these basic principles, you can tackle a wide range of algebraic equations with confidence.

Furthermore, remember that fractions and decimals are just different ways of representing the same number. Being comfortable converting between fractions (like 7/2) and decimals (like 3.5) can be helpful in various mathematical contexts. It’s a good practice to familiarize yourself with these conversions and understand when one form might be more convenient than the other.

Simplifying √{89 - 28√10}

Now, let's move on to the second part of our challenge: simplifying the expression √{89 - 28√10}. This looks a bit more complex, but don't worry, we can handle it! The key here is to recognize that the expression inside the square root might be a perfect square in disguise. We need to manipulate it to reveal that hidden perfect square.

When we see a square root containing another square root, like √{89 - 28√10}, it often suggests that we're dealing with an expression of the form (a - b√c)². If we can rewrite 89 - 28√10 in this form, then taking the square root becomes much simpler.

Let's assume that √{89 - 28√10} can be expressed as (a - b√10) for some integers a and b. Squaring this expression gives us:

(a - b√10)² = a² - 2ab√10 + (b√10)² = a² - 2ab√10 + 10b²

We want this to be equal to 89 - 28√10. So, we can set up a system of equations by equating the coefficients of the terms without the square root and the terms with the square root:

  1. a² + 10b² = 89
  2. -2ab = -28

From the second equation, we can simplify it to ab = 14. Now we need to find integer values for a and b that satisfy both equations. Since ab = 14, the possible pairs of (a, b) are (1, 14), (2, 7), (7, 2), and (14, 1). Let's test these pairs in the first equation:

  • If (a, b) = (1, 14), a² + 10b² = 1 + 10(196) = 1961 (too large)
  • If (a, b) = (2, 7), a² + 10b² = 4 + 10(49) = 494 (too large)
  • If (a, b) = (7, 2), a² + 10b² = 49 + 10(4) = 49 + 40 = 89 (This works!)
  • If (a, b) = (14, 1), a² + 10b² = 196 + 10 = 206 (too large)

So, the pair (a, b) = (7, 2) satisfies both equations. This means that 89 - 28√10 = (7 - 2√10)². Therefore, √{89 - 28√10} = √{(7 - 2√10)²}. Taking the square root of a square gives us the absolute value of the expression, so:

√{89 - 28√10} = |7 - 2√10|

Since 2√10 is approximately 2 * 3.16 = 6.32, which is less than 7, 7 - 2√10 is positive. Therefore, we can drop the absolute value signs:

√{89 - 28√10} = 7 - 2√10

So, the simplified form of √{89 - 28√10} is 7 - 2√10. That's a pretty neat result, huh?

Deeper Dive into Simplifying Nested Radicals

Simplifying nested radicals like √{89 - 28√10} can seem intimidating at first, but the underlying principle is quite elegant. We're essentially trying to reverse the process of squaring a binomial that involves a radical. The general form we look for is (a ± b√c)², where a, b, and c are integers.

When you square a binomial like this, you get:

(a ± b√c)² = a² ± 2ab√c + b²c

Notice how the result has two parts: a term without a radical (a² + b²c) and a term with a radical (± 2ab√c). This is the key to our simplification strategy. We want to match these parts with the given expression under the nested radical.

In our example, we had √{89 - 28√10}. We identified that 89 corresponds to a² + 10b² (since c = 10 in this case) and -28√10 corresponds to -2ab√10. By setting up the system of equations, we could solve for a and b. This approach allows us to "unravel" the nested radical and express it in a simpler form.

This technique is not just a mathematical trick; it highlights the interconnectedness of different mathematical concepts. It combines algebra (manipulating equations), number theory (finding integer solutions), and an understanding of radicals and binomial expansion. Mastering these types of problems strengthens your overall mathematical toolkit and enhances your problem-solving skills.

Conclusion

So, there you have it! We successfully solved the equation -x = -7/2, finding that x equals 7/2 or 3.5. We also simplified the complex-looking radical expression √{89 - 28√10}, which turned out to be 7 - 2√10. These problems might seem daunting at first glance, but with a systematic approach and a solid understanding of the underlying principles, they become much more manageable. Keep practicing, and you'll be a math whiz in no time! Remember, math is like a puzzle – sometimes you need to try different approaches, but the satisfaction of solving it is totally worth it. Keep up the great work, guys!