Let's dive into solving for alpha (α) in these two equations. We'll break down each equation step by step to make it super easy to follow. Grab your thinking caps, guys, because we're about to do some math!

Equation 1: α/2 = 2/α

Understanding the Equation

So, we're given that α/2 is equal to 2/α. What does this mean? It means that some number, alpha, divided by 2, gives the same result as 2 divided by that same number. Our mission is to find out what this mysterious number alpha is.

Cross-Multiplication

The easiest way to tackle this equation is by using cross-multiplication. Cross-multiplication is a handy technique when you have two fractions equal to each other. You multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa.

In our case, we'll multiply α (the numerator of the first fraction) by α (the denominator of the second fraction), and we'll multiply 2 (the denominator of the first fraction) by 2 (the numerator of the second fraction). This gives us:

α * α = 2 * 2

Which simplifies to:

α² = 4

Solving for α

Now we have α² = 4. To find α, we need to take the square root of both sides of the equation. Remember, when you take the square root, you get two possible solutions: a positive and a negative value.

So, taking the square root of both sides gives us:

α = ±√4

Which means:

α = 2 or α = -2

Two Possible Solutions

So far, we have two possible solutions for α based on the first equation: 2 and -2. Keep these in mind as we move on to the second equation. This step is crucial to ensure we consider all possibilities and don't miss any potential answers. Understanding that α can be either 2 or -2 sets the stage for further analysis, making sure we're thorough in our problem-solving approach. Remember, math is all about precision and attention to detail!

Equation 2: α³ = 125

Understanding the Equation

Now let's look at the second equation: α³ = 125. This equation tells us that alpha, when multiplied by itself three times (cubed), equals 125. We need to find the value of alpha that satisfies this condition.

Finding the Cube Root

To find α, we need to take the cube root of both sides of the equation. The cube root of a number is the value that, when multiplied by itself three times, gives you the original number. In mathematical terms:

α = ∛125

Calculating the Cube Root of 125

What number, when multiplied by itself three times, equals 125? Well, 5 * 5 * 5 = 125. Therefore, the cube root of 125 is 5.

α = 5

Unique Solution

Unlike the square root, the cube root of a positive number gives you only one real solution. So, from the second equation, we find that α = 5. This simplifies our possibilities and allows us to focus on a single, concrete value. Understanding this difference between square roots and cube roots is essential for solving equations accurately. By recognizing that α must be 5, we're one step closer to the final answer.

Combining the Results

Comparing Solutions

Okay, so from the first equation (α/2 = 2/α), we found that α could be either 2 or -2. From the second equation (α³ = 125), we found that α must be 5.

Identifying the Correct Value

Now, we need to find a value of α that satisfies both equations. Looking at our solutions, we see that α = 5 is the only value that works for the second equation. However, it doesn't satisfy the first equation. On the other hand, α = 2 and α = -2 satisfy the first equation but not the second.

Checking α = 5 in the First Equation

Let's plug α = 5 into the first equation to see if it holds true:

5/2 = 2/5

2. 5 = 0.4

This is clearly not true, so α = 5 does not satisfy the first equation.

Checking α = 2 and α = -2 in the Second Equation

Let's check α = 2:

2³ = 2 * 2 * 2 = 8

8 ≠ 125

Now let's check α = -2:

(-2)³ = -2 * -2 * -2 = -8

-8 ≠ 125

Neither 2 nor -2 satisfy the second equation. This detailed checking process ensures we don't make any assumptions and verify each solution against the original equations. The goal here is to find a value of α that consistently works across all given conditions.

The Importance of Verification

It's super important to verify solutions! Always plug your potential answers back into the original equations to make sure they work. This step can save you from making mistakes and helps you catch any errors in your calculations. Verifying solutions is like double-checking your work; it ensures you're on the right track and haven't overlooked anything. It's a habit that builds confidence in your answers and reinforces your understanding of the problem-solving process.

Conclusion

After analyzing both equations, we find that there is no value of α that satisfies both α/2 = 2/α and α³ = 125 simultaneously. The solutions derived from each equation are mutually exclusive. Therefore, there is no single value for alpha that makes both equations true at the same time. This kind of problem highlights the importance of checking all conditions and ensuring consistency across multiple equations. Sometimes, the answer isn't a specific number but rather the realization that no solution exists that fits all the criteria.