Solving For M: A Radical Equation Challenge!

Hey guys! Let's dive into a fun math problem together. We've got a radical equation, and our mission is to find the value of 'm' and then figure out what 4+2√3 times 'm' equals. Buckle up, it's gonna be a ride!

Understanding the Problem

So, the problem states that m is greater than 1, and we have this equation: √√3m-m=√√3-1. Our ultimate goal is to find the value of m and then determine what 4+2√3 times m is. The possible answers are A) 1, B) √2, C) 3/2, D) √3, and E) 4. Let's break it down step by step to make sure we get to the right answer.

Rewriting the Equation

The first thing we need to do is rewrite the equation to make it easier to work with. The given equation is √√3m-m=√√3-1. Notice that we can factor out 'm' from the left side of the equation. This gives us:

√[m(√3-1)] = √√3-1

This looks a bit cleaner, right? Now we can see that 'm' is multiplied by (√3-1) inside the outer square root.

Squaring Both Sides (Twice!)

To get rid of the square roots, we're going to have to square both sides of the equation... twice! First, let's square both sides once:

[√[m(√3-1)]]^2 = [√√3-1]^2

This simplifies to:

√(m(√3-1)) = √3-1

Okay, we still have a square root, so let's square both sides again:

[√(m(√3-1))]^2 = (√3-1)^2

This gives us:

m(√3-1) = (√3-1)^2

Isolating 'm'

Now we want to isolate 'm'. To do this, we divide both sides of the equation by (√3-1):

m = (√3-1)^2 / (√3-1)

Since we're dividing (√3-1)^2 by (√3-1), we can simplify this to:

m = √3-1

So, we've found that m = √3-1.

Finding (4+2√3) times 'm'

Now that we know the value of 'm', we can find the value of (4+2√3) times 'm'. Let's plug in the value of 'm':

(4+2√3) * m = (4+2√3) * (√3-1)

Expanding the Expression

To simplify this, we need to expand the expression. We'll use the distributive property (also known as FOIL - First, Outer, Inner, Last):

(4+2√3) * (√3-1) = 4*√3 - 41 + 2√3√3 - 2√3*1

This simplifies to:

4√3 - 4 + 2*3 - 2√3

Which further simplifies to:

4√3 - 4 + 6 - 2√3

Combining Like Terms

Now, let's combine the like terms (the terms with √3 and the constants):

(4√3 - 2√3) + (-4 + 6)

This gives us:

2√3 + 2

Factoring Out a 2

We can factor out a 2 from the expression:

2(√3 + 1)

But wait! This doesn't match any of the answer choices. Let's go back and check our work to see if we made a mistake.

Spotting the Mistake!

Okay, after reviewing our steps, we spot a crucial detail we missed early on! Remember when we had:

√[m(√3-1)] = √√3-1

We incorrectly simplified the right side when squaring. The correct initial equation to work with after the first square is:

√(m(√3 - 1)) = √(√3 - 1)

When we square both sides the second time, we should have:

m(√3 - 1) = √3 - 1

Dividing both sides by (√3 - 1) gives us:

m = (√3 - 1) / (√3 - 1)

Which simplifies to:

m = 1

Recalculating (4+2√3) times 'm' with the Corrected 'm'

Now that we have the correct value for 'm', which is m = 1, we can easily find the value of (4+2√3) times 'm':

(4 + 2√3) * m = (4 + 2√3) * 1

This simplifies to:

4 + 2√3

But we still need to figure out what (4 + 2√3) times 'm' equals based on the given options. Since m = 1, the expression (4 + 2√3) * 1 is simply (4 + 2√3). We made another mistake in the original problem setup and how it affects the answer options.

Correcting the Final Step

Since m = 1, the question asks for the value of (4 + 2√3) * m = (4 + 2√3) * 1, which simplifies to 4 + 2√3. Now let's check the answer options again and see what the question intended for us to solve for.

If the question meant "what value, when multiplied by m, equals 4 + 2√3", and given the answer choices, we need to see if any of the options, when multiplied by our corrected m = 1, would result in one of the provided answer choices. Since m = 1:

(4 + 2√3) * m = 4 + 2√3.

Upon closer inspection, there was an error in the initial setup. The equation should have led to m = √3 - 1. Now, let’s correct the final step with m = √3 - 1.

(4 + 2√3) * m = (4 + 2√3) * (√3 - 1)

= 4√3 - 4 + 2 * 3 - 2√3

= 4√3 - 4 + 6 - 2√3

= 2√3 + 2

= 2(√3 + 1)

Still, this doesn't directly match any of the answer choices provided. Let's carefully re-examine the original equation and each step to pinpoint where the discrepancy lies.

Final Solution

After a meticulous review, the original steps are correct up to the point where we find m = √3 - 1. The issue arises in the interpretation of the final question. We need to find what (4 + 2√3) * m equals:

(4 + 2√3) * (√3 - 1) = 4√3 - 4 + 6 - 2√3 = 2√3 + 2 = 2(1 + √3)

None of the provided answer choices directly match 2(1 + √3). However, if the question implicitly asked, "What is m * (4 + 2√3) simplified, and is it equal to any of the given options if something is off by a constant factor?"

The most likely correct solution is obtained if the question intended us to find a factor close to 4+2√3.

Given m = √3-1. Let's rationalize the term to relate the option, if applicable.

If the question's answers are incorrect, the closest might relate back to the initial terms.

So we need to find (4 + 2√3) * m = (4 + 2√3)(√3 - 1) = 2√3 + 2, which still does not relate. Therefore, based on the original problem and the correct execution of steps, none of the provided answer choices (A) 1, (B) √2, (C) 3/2, (D) √3, (E) 4 appear to be correct. There might be an issue with the original problem statement or the answer choices.

Therefore, since the correct and simplified result (2√3 + 2) does not align with any provided options, we cannot select one. It's essential to double-check the original question or answer options for accuracy.