Solving For X: 45:[129:3-3x(11+3x)+4]=9

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Alright guys, let's dive into solving this math problem together! We've got the equation 45:[129:3-3x(11+3x)+4]=9, and our mission is to find the natural number x that makes this equation true. It looks a bit intimidating at first, but don't worry, we'll break it down step by step.

Understanding the Equation

First, let's make sure we understand what the equation is telling us. We have a series of operations involving x, and we need to figure out what value of x will make the whole thing equal to 9. The key here is to follow the order of operations (PEMDAS/BODMAS) and simplify the expression inside the brackets first. The natural number x we are looking for must satisfy the given equation, meaning when we substitute that value into the equation, both sides will be equal. Our goal is to isolate x and find its value, but to do that, we'll need to simplify the equation systematically.

We need to carefully analyze each part of the equation. The natural number x is hidden within a complex expression, so we'll start by simplifying the innermost parts first. We'll deal with the parentheses, then multiplication, division, addition, and subtraction, in that order, always working from the inside out. This systematic approach will help us avoid errors and keep track of our progress. Remember, every step we take is a step closer to finding the value of x.

To further grasp this, let's take a closer look at the different components of the equation. We have division, multiplication, addition, and subtraction, all intertwined. The brackets and parentheses act as containers, grouping operations together. This means we have to tackle the innermost groupings first before we can simplify the outer layers. This is like peeling an onion, we need to peel each layer to get to the center. Understanding this structure is crucial for correctly solving the equation. The natural number x is our target, but we need to navigate through this complex mathematical landscape to find it.

Step-by-Step Solution

  1. Isolate the bracketed expression: To start, let's isolate the bracketed expression. We can rewrite the equation as: 45 / [129/3 - 3x(11 + 3x) + 4] = 9 To get rid of the division, we can multiply both sides by the bracketed expression: 45 = 9 * [129/3 - 3x(11 + 3x) + 4] Now, divide both sides by 9: 5 = 129/3 - 3x(11 + 3x) + 4 This simplifies things quite a bit, making it easier to work with the expression inside the brackets.

  2. Simplify the division: Next, let's simplify the division 129/3: 129 / 3 = 43 So our equation now looks like: 5 = 43 - 3x(11 + 3x) + 4 We're making progress! By simplifying the division, we've reduced the complexity of the equation and brought us closer to isolating the terms with x.

  3. Combine constants: Now, let's combine the constant terms 43 and 4: 43 + 4 = 47 Our equation becomes: 5 = 47 - 3x(11 + 3x) Combining constants helps to clean up the equation and makes it easier to see the relationship between the terms involving x and the constant terms. This step is crucial for moving towards isolating x.

  4. Isolate the term with x: Let's isolate the term with x by subtracting 47 from both sides: 5 - 47 = -3x(11 + 3x) -42 = -3x(11 + 3x) This step is crucial because it separates the terms involving x from the constants, allowing us to focus on simplifying the expression with x.

  5. Divide by -3: Divide both sides by -3: -42 / -3 = x(11 + 3x) 14 = x(11 + 3x) Dividing by -3 simplifies the equation further, making it easier to expand and rearrange the terms. This step is essential for revealing the quadratic nature of the equation.

  6. Expand the expression: Now, expand the expression on the right side: 14 = 11x + 3x^2 Expanding the expression allows us to see the equation in a standard quadratic form, which is necessary for solving for x.

  7. Rearrange into a quadratic equation: Rearrange the equation into the standard quadratic form (ax^2 + bx + c = 0): 3x^2 + 11x - 14 = 0 Rearranging the equation into standard quadratic form is a critical step because it allows us to use standard methods, such as factoring or the quadratic formula, to solve for x.

  8. Solve the quadratic equation: We can solve this quadratic equation by factoring. We're looking for two numbers that multiply to (3 * -14 = -42) and add up to 11. Those numbers are 14 and -3. Rewrite the middle term using these numbers: 3x^2 + 14x - 3x - 14 = 0 Now, factor by grouping: x(3x + 14) - 1(3x + 14) = 0 (x - 1)(3x + 14) = 0 This gives us two possible solutions for x: x - 1 = 0 => x = 1 3x + 14 = 0 => x = -14/3 Factoring the quadratic equation is a crucial step in finding the possible values of x. By expressing the quadratic as a product of two binomials, we can easily identify the values of x that make the equation true.

  9. Check for natural number solutions: Since we're looking for a natural number solution, we can discard x = -14/3 because it's not a natural number. A natural number x must be a positive integer. So, the solution is: x = 1 Checking for natural number solutions is essential because the problem specifically asks for a natural number. This step involves verifying that the solutions obtained meet the criteria specified in the problem statement.

Final Answer

Therefore, the natural number x that satisfies the equation 45:[129:3-3x(11+3x)+4]=9 is x = 1. We have successfully navigated through the complexities of the equation and found our answer. Remember, the key is to break down the problem into smaller, manageable steps and to stay organized throughout the process. The natural number x we found makes the equation balance perfectly. Great job, guys!