Easy Math: Solve 2m² = 8

Let's dive into a straightforward math problem that often pops up in algebra: solving for a variable when it's squared. We're looking at the equation $2m^2 = 8$. Don't let the squared term intimidate you; we'll break it down step-by-step so you can see just how simple it is to find the value(s) of 'm'. Whether you're a student tackling homework or just enjoy flexing your mental math muscles, understanding how to isolate and solve for a squared variable is a fundamental skill.

Understanding the Equation

First things first, let's look at the equation $2m^2 = 8$. Our goal here is to find out what number(s) 'm' can be to make this statement true. The 'm' is being multiplied by itself (that's what the square, $^2$, means) and then that result is being multiplied by 2. All of this equals 8. To solve for 'm', we need to work backward, essentially undoing the operations that are being applied to it. Think of it like unwrapping a present – you take off the outer layers first to get to the gift inside. In this case, the 'gift' is the value of 'm', and the 'layers' are the multiplication by 2 and the squaring.

Step 1: Isolate the Squared Term

The first step in solving $2m^2 = 8$ is to get the $m^2$ term all by itself on one side of the equation. Right now, it's being multiplied by 2. To undo multiplication, we use division. So, we're going to divide both sides of the equation by 2. This is crucial because we want to isolate the variable part ($m^2$) before we deal with the square itself. By dividing both sides by 2, we maintain the equality of the equation. Remember, whatever you do to one side of an equation, you must do to the other side to keep it balanced. So, we have:

$2m^2 / 2 = 8 / 2$

This simplifies to:

$m^2 = 4$

See? We've successfully isolated $m^2$. It now stands alone, making it easier to tackle the next step. This might seem like a small move, but it's a vital one in solving any equation where a variable is squared. Always aim to get that squared term by itself first!

Step 2: Solve for 'm'

Now that we have $m^2 = 4$, we need to find out what 'm' is. The square root operation is the inverse of squaring. When you square a number, you multiply it by itself (e.g., $3^2 = 3 * 3 = 9$). To find the original number that was squared, you take the square root. So, we need to take the square root of both sides of our equation:

$\sqrt{m^2} = \sqrt{4}$

When we take the square root of $m^2$, we get 'm'. Now, let's think about the square root of 4. What number, when multiplied by itself, equals 4? Well, $2 * 2 = 4$. So, 'm' could be 2. However, here's a critical point in mathematics: negative numbers also produce positive results when squared. For example, $(-2) * (-2) = 4$ as well! This means that when we take the square root of a number in an equation like this, there are two possible answers: a positive one and a negative one.

Therefore, the square root of 4 is not just 2, but also -2. This is why we use the 'plus-minus' symbol ($\pm$) to represent both possibilities. So, the solution is:

$m = \pm 2$

This indicates that both $m = 2$ and $m = -2$ are valid solutions to the original equation $2m^2 = 8$. It's essential to remember this when dealing with squared variables, as you often need to consider both the positive and negative roots.

Verifying the Solutions

To be absolutely sure, let's check our answers by plugging them back into the original equation, $2m^2 = 8$.

• Checking $m = 2$: Substitute 2 for 'm': $2 * (2)^2 = 2 * 4 = 8$. This is correct!

• Checking $m = -2$: Substitute -2 for 'm': $2 * (-2)^2 = 2 * 4 = 8$. This is also correct!

Since both $m = 2$ and $m = -2$ satisfy the equation, our solution $m = \pm 2$ is accurate.

Conclusion

Solving the equation $2m^2 = 8$ led us to the answer $m = \pm 2$. We achieved this by first isolating the $m^2$ term through division and then finding the square root of both sides, remembering to account for both positive and negative possibilities. This process highlights a fundamental algebraic technique applicable to many similar problems. For further exploration into algebraic equations and problem-solving techniques, you might find resources from mathsisfun.com very helpful.

Answer Choices

Given the options:

A. $m=8$ B. $m= extrm{±} 4$ C. $m= extrm{±} 2$ D. $m=2$

Our calculated solution, $m = \pm 2$, directly matches option C.