Simplifying The Equation: 234 × (2x + 2 × 2x + 3) = 23x

October 11, 2025 by TextBrain Team 56 views

Hey guys! Let's dive into simplifying this equation: 234 × (2x + 2 × 2x + 3) = 23x [2 × +2 + x + 3]. Math can seem daunting, but breaking it down step by step makes it totally manageable. We’ll go through each part, making sure we understand what’s happening. So grab your thinking caps, and let’s get started!

Understanding the Equation

Before we jump into solving, let's take a good look at our equation: 234 × (2x + 2 × 2x + 3) = 23x [2 × +2 + x + 3]. The first thing we notice is that there are terms involving 'x', constants, and multiplication. Our goal is to simplify both sides of the equation and then isolate 'x' to find its value. This might involve distributing numbers, combining like terms, and rearranging the equation.

When you're faced with an equation like this, always start by identifying the operations and the order in which they need to be performed. Remember PEMDAS/BODMAS: Parentheses/Brackets first, then Exponents/Orders, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). By following this order, we can ensure we simplify each side correctly.

Another crucial aspect is to keep track of each step. Write everything down clearly so you don't lose sight of what you've done. This helps in avoiding mistakes and makes it easier to check your work later. Equations like these can be intricate, so a systematic approach is your best friend. We're aiming to make the equation cleaner and more straightforward, making it easier to solve for 'x'.

Breaking Down the Left-Hand Side

Let's tackle the left-hand side (LHS) of the equation: 234 × (2x + 2 × 2x + 3). We'll use the order of operations (PEMDAS/BODMAS) to guide us. First up are the operations within the parentheses. Inside the parentheses, we have addition and multiplication. Multiplication comes before addition, so we'll address the term 2 × 2x first. This simplifies to 4x. Now our expression inside the parentheses looks like this: 2x + 4x + 3.

Next, we combine the like terms, which are the terms involving 'x'. We have 2x and 4x, which add up to 6x. So now the expression inside the parentheses is 6x + 3. This simplifies our LHS to 234 × (6x + 3). The next step is to distribute the 234 across the terms inside the parentheses. This means multiplying 234 by both 6x and 3. So, 234 × 6x equals 1404x, and 234 × 3 equals 702. Thus, the simplified LHS is 1404x + 702.

Breaking down the LHS step-by-step like this ensures we don't miss any details. Each step is clear and logical, making it easier to follow along and reducing the chances of error. Remember, the key to simplifying complex expressions is to take it one operation at a time, always following the correct order. We've now got the LHS simplified to a more manageable form, and we're ready to move on to the right-hand side.

Simplifying the Right-Hand Side

Now, let's simplify the right-hand side (RHS) of the equation: 23x [2 × +2 + x + 3]. Just like we did with the LHS, we need to follow the order of operations (PEMDAS/BODMAS) to make sure we simplify correctly. First, let's address the expression inside the brackets: [2 × +2 + x + 3]. It looks like there might be a typo in this expression, but we'll interpret 2 × +2 as simply 2 + 2, which equals 4. So, our expression inside the brackets becomes [4 + x + 3].

Next, we combine the constants 4 and 3, which gives us 7. Now, the expression inside the brackets is [x + 7]. So, the RHS of the equation now looks like this: 23x (x + 7). To further simplify, we need to distribute the 23x across the terms inside the parentheses. This means multiplying 23x by both x and 7. So, 23x × x equals 23x², and 23x × 7 equals 161x. Therefore, the simplified RHS is 23x² + 161x.

By carefully breaking down the RHS and dealing with each term step-by-step, we've managed to simplify it significantly. We identified the operations, handled the constants, and distributed the terms as needed. This approach helps keep everything clear and reduces the risk of making mistakes. Now that we've simplified both the LHS and RHS, we're ready to bring them together and solve for 'x'.

Combining and Rearranging the Equation

With the left-hand side (LHS) simplified to 1404x + 702 and the right-hand side (RHS) simplified to 23x² + 161x, we can now combine the two sides to form the complete equation: 1404x + 702 = 23x² + 161x. To solve for 'x', we need to rearrange the equation so that all terms are on one side, setting the equation to zero. This will give us a quadratic equation that we can solve using various methods.

Let's start by subtracting 1404x and 702 from both sides of the equation. This gives us: 0 = 23x² + 161x - 1404x - 702. Next, we combine the like terms involving 'x': 161x - 1404x = -1243x. So, our equation now looks like this: 0 = 23x² - 1243x - 702. This is a quadratic equation in the form of ax² + bx + c = 0, where a = 23, b = -1243, and c = -702.

Rearranging the equation into a standard quadratic form is a crucial step because it allows us to apply methods like factoring, completing the square, or the quadratic formula to find the values of 'x'. By setting the equation to zero and aligning the terms, we've set the stage for the final steps in solving for 'x'.

Solving for 'x'

Now that we have the quadratic equation 0 = 23x² - 1243x - 702, we need to solve for 'x'. There are several methods we could use, including factoring, completing the square, or applying the quadratic formula. Given the coefficients, factoring might be challenging, and completing the square can be cumbersome. So, let's use the quadratic formula, which is a reliable method for solving any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is:

x = [ -b ± √(b² - 4ac) ] / (2a)

In our equation, a = 23, b = -1243, and c = -702. Plugging these values into the quadratic formula, we get:

x = [ -(-1243) ± √((-1243)² - 4 × 23 × (-702)) ] / (2 × 23)

First, let's simplify the expression under the square root:

(-1243)² = 1545049
4 × 23 × (-702) = -64584
So, 1545049 - (-64584) = 1545049 + 64584 = 1609633

Now our equation looks like this:

x = [ 1243 ± √1609633 ] / 46

Next, we calculate the square root of 1609633, which is approximately 1268.71. So, we have:

x = [ 1243 ± 1268.71 ] / 46

This gives us two possible values for 'x':

x = (1243 + 1268.71) / 46 = 2511.71 / 46 ≈ 54.6
x = (1243 - 1268.71) / 46 = -25.71 / 46 ≈ -0.56

So, the solutions for 'x' are approximately 54.6 and -0.56. Using the quadratic formula allows us to methodically find the solutions, even when dealing with large numbers and complex equations. We've now successfully solved for 'x', and we're at the final step of verifying our solution.

Verifying the Solution

After solving for 'x', it's super important to verify the solutions to make sure they're correct. We found two possible values for 'x': approximately 54.6 and -0.56. To verify, we'll plug each value back into the original equation: 234 × (2x + 2 × 2x + 3) = 23x [2 × +2 + x + 3]. Let's start with x ≈ 54.6.

Plugging x = 54.6 into the equation, we need to evaluate both the left-hand side (LHS) and the right-hand side (RHS) separately and see if they are equal.

LHS: 234 × (2(54.6) + 2 × 2(54.6) + 3)

2(54.6) = 109.2
2 × 2(54.6) = 4(54.6) = 218.4
So, the expression inside the parentheses becomes: 109.2 + 218.4 + 3 = 330.6
LHS = 234 × 330.6 ≈ 77360.4

RHS: 23(54.6) [2 × +2 + 54.6 + 3]

We interpreted 2 × +2 as 4, so the expression inside the brackets becomes: [4 + 54.6 + 3] = 61.6

23(54.6) = 1255.8
RHS = 1255.8 × 61.6 ≈ 77356.1

The LHS and RHS are approximately equal (77360.4 ≈ 77356.1), so x ≈ 54.6 seems to be a valid solution. Now, let's verify the other solution, x ≈ -0.56.

Plugging x = -0.56 into the equation:

LHS: 234 × (2(-0.56) + 2 × 2(-0.56) + 3)

2(-0.56) = -1.12
2 × 2(-0.56) = 4(-0.56) = -2.24
So, the expression inside the parentheses becomes: -1.12 - 2.24 + 3 = -0.36
LHS = 234 × (-0.36) ≈ -84.24

RHS: 23(-0.56) [2 × +2 + (-0.56) + 3]

Again, we interpreted 2 × +2 as 4, so the expression inside the brackets becomes: [4 - 0.56 + 3] = 6.44

23(-0.56) = -12.88
RHS = -12.88 × 6.44 ≈ -82.94

The LHS and RHS are approximately equal (-84.24 ≈ -82.94), so x ≈ -0.56 also appears to be a valid solution. By verifying both solutions, we increase our confidence in the accuracy of our results. This step is crucial to ensure that we haven't made any mistakes in our calculations. Remember, guys, always verify your solutions!

Conclusion

Alright, guys! We did it! We successfully simplified and solved the equation 234 × (2x + 2 × 2x + 3) = 23x [2 × +2 + x + 3]. We started by breaking down the equation step by step, simplifying both the left-hand side (LHS) and the right-hand side (RHS) following the order of operations (PEMDAS/BODMAS). Then, we combined the simplified sides, rearranged the equation into a quadratic form, and used the quadratic formula to find the solutions for 'x'.

We found two possible solutions: x ≈ 54.6 and x ≈ -0.56. To ensure accuracy, we verified both solutions by plugging them back into the original equation. Both values made the equation approximately true, giving us confidence in our results. Remember, guys, the key to solving complex equations is to take it one step at a time, be organized, and double-check your work.

Math might seem intimidating sometimes, but with a systematic approach and a bit of practice, you can tackle even the trickiest problems. Keep practicing, stay curious, and don't be afraid to break down problems into smaller, more manageable steps. You got this! Now go out there and conquer those equations! Thanks for joining me on this mathematical journey!