Positive Real Number Find The Solution To The Inverse Subtraction Puzzle

Hey there, math enthusiasts! Ever stumbled upon a math problem that seems like a riddle wrapped in an equation? Today, we're diving headfirst into one of those intriguing puzzles. We're going to explore how to find a positive real number that, when you subtract its inverse, gives you exactly 2. Sounds like fun, right? So, let's put on our thinking caps and get started!

Understanding the Problem

Before we jump into solving, let's break down what the question is really asking. We're looking for a number, let's call it 'x', that fits this equation:

x - (1/x) = 2

In simpler terms, we need to find a number where subtracting its reciprocal (1 divided by the number) equals 2. This isn't your typical straightforward math problem; it requires a bit of algebraic maneuvering and a dash of logical thinking. But don't worry, we'll walk through it step by step. To help us out, we have a few answer choices: a) 3, b) 4, c) 5, and d) 6. This means we can either try to solve the equation directly or test each option to see which one fits. Both methods are perfectly valid, and we'll explore both to give you a comprehensive understanding.

Setting Up the Algebraic Equation

To solve this algebraically, our first step is to get rid of the fraction. We can do this by multiplying every term in the equation by 'x'. This gives us:

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

This simplifies to:

x² - 1 = 2x

Now, we have a quadratic equation! To solve it, we need to rearrange the equation so that it equals zero. Subtracting 2x from both sides, we get:

x² - 2x - 1 = 0

Solving the Quadratic Equation

Okay, we've got a quadratic equation in the standard form (ax² + bx + c = 0). There are a couple of ways we can solve this: factoring or using the quadratic formula. Factoring can be tricky if the equation doesn't have nice, whole-number roots. In this case, the quadratic formula is our best bet. Remember the quadratic formula? It's:

x = (-b ± √(b² - 4ac)) / 2a

Where 'a', 'b', and 'c' are the coefficients from our quadratic equation (x² - 2x - 1 = 0). So, in our equation:

• a = 1
• b = -2
• c = -1

Let's plug these values into the quadratic formula:

x = (2 ± √((-2)² - 4 * 1 * -1)) / (2 * 1)

Simplifying this, we get:

x = (2 ± √(4 + 4)) / 2

x = (2 ± √8) / 2

x = (2 ± 2√2) / 2

Now, we can simplify further by dividing both terms in the numerator by 2:

x = 1 ± √2

This gives us two possible solutions:

• x = 1 + √2
• x = 1 - √2

Choosing the Correct Solution

Remember, the question specifically asks for a positive real number. Let's look at our two solutions:

• x = 1 + √2 (approximately 1 + 1.414 = 2.414) This is a positive number.
• x = 1 - √2 (approximately 1 - 1.414 = -0.414) This is a negative number.

Since we're looking for a positive solution, the answer is x = 1 + √2. However, this answer isn't one of our multiple-choice options (a) 3, b) 4, c) 5, and d) 6). This indicates that we might need to rethink our approach slightly or that the provided options are not directly derived from the exact solution, but one of the options might be the correct answer.

Answering by Checking Alternatives

Given that the exact solution (1 + √2) isn't among the options, it's very likely there was a mistake in the question, or the expected answer format is different. If we still need to choose from the provided alternatives, the correct approach is to verify each alternative in the original equation to see which one satisfies it.

The original equation is:

x - (1/x) = 2

We'll replace 'x' with each of the given options and check if the equation holds true.

Verifying Option A) x = 3

Replace x with 3:

3 - (1/3) = 2

Convert all terms to have a common denominator to compare:

(9/3) - (1/3) = 2

(8/3) = 2

8/3 is not equal to 2, so option A is incorrect.

Verifying Option B) x = 4

Replace x with 4:

4 - (1/4) = 2

Convert all terms to have a common denominator to compare:

(16/4) - (1/4) = 2

(15/4) = 2

15/4 is not equal to 2, so option B is incorrect.

Verifying Option C) x = 5

Replace x with 5:

5 - (1/5) = 2

Convert all terms to have a common denominator to compare:

(25/5) - (1/5) = 2

(24/5) = 2

24/5 is not equal to 2, so option C is incorrect.

Verifying Option D) x = 6

Replace x with 6:

6 - (1/6) = 2

Convert all terms to have a common denominator to compare:

(36/6) - (1/6) = 2

(35/6) = 2

35/6 is not equal to 2, so option D is incorrect.

Conclusion of Alternatives Checking

After checking each alternative in the original equation, we found that none of the options (3, 4, 5, or 6) satisfy the equation x - (1/x) = 2. This strongly suggests that there might be an issue with the question itself or the provided alternatives, as none of them fit the condition.

Method 2: Testing the Options

Since we have multiple-choice options, let's try plugging each one into our original equation and see if it works. This method can be quicker than solving the quadratic equation, especially if one of the options jumps out as the answer.

Testing Option A: x = 3

Let's substitute x = 3 into the equation:

3 - (1/3) = 2

Simplifying, we get:

3 - 0.333... = 2

2. 666... = 2

This is not true, so option A is incorrect.

Testing Option B: x = 4

Now, let's try x = 4:

4 - (1/4) = 2

Simplifying:

4 - 0.25 = 2

3. 75 = 2

This is also not true, so option B is incorrect.

Testing Option C: x = 5

Let's move on to x = 5:

5 - (1/5) = 2

Simplifying:

5 - 0.2 = 2

4. 8 = 2

Again, this is not true, so option C is incorrect.

Testing Option D: x = 6

Finally, let's test x = 6:

6 - (1/6) = 2

Simplifying:

6 - 0.1666... = 2

5. 833... = 2

This is not true either, so option D is incorrect.

Analysis of Alternative Tests

After individually testing each alternative (3, 4, 5, and 6) in the equation x - (1/x) = 2, we've determined that none of them satisfy the equation. This outcome suggests a discrepancy, possibly indicating an error in either the problem formulation or the provided options. In practical scenarios, such a result prompts a re-evaluation of the original question and the options to ensure accuracy.

Final Thoughts and Problem Reassessment

So, guys, we've tackled this math puzzle from both angles. We solved the equation algebraically and tested each of the given options. Interestingly, neither method gave us a direct match within the provided choices. This usually means one of two things: either there's a slight error in the question itself, or the answer might be looking for something a bit different than we initially expected.

Perhaps the question intended to ask for an approximate solution, or maybe there was a typo in the options. In real-world problem-solving, this is a valuable lesson: sometimes, the most important step is to double-check the problem itself. Math isn't just about finding the right answer; it's also about understanding the question and making sure it makes sense.

If we were to approximate, we know the correct answer is 1 + √2, which is roughly 2.414. None of the options are close to this value. It’s important to always reflect on your solution and consider whether it aligns with the context of the problem. In this case, the discrepancy between our solution and the options provided highlights the importance of critical thinking in mathematics.

Keep practicing, keep questioning, and remember that every puzzle, even the tricky ones, helps us sharpen our minds. Until next time, happy problem-solving!