Solved: (x+4)³-(x-3)³=343 Step-by-Step Solution

by Ahmed Latif 48 views

Hey there, math enthusiasts! Today, we're diving into an intriguing algebraic equation: (x+4)³ - (x-3)³ = 343. This might look daunting at first glance, but don't worry, we'll break it down step-by-step, making it super easy to understand. So, grab your calculators, put on your thinking caps, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the equation is asking us. We have two cubic expressions, (x+4)³ and (x-3)³, and we're subtracting the second from the first. The result of this subtraction is equal to 343. Our goal is to find the value(s) of 'x' that make this equation true. This involves some algebraic manipulation and a bit of clever thinking. We need to expand the cubic expressions, simplify the equation, and then solve for 'x'. There are a couple of ways we can approach this problem. One way is to expand the cubes directly using the binomial theorem or by manually multiplying the expressions. Another way is to use the identity a³ - b³ = (a - b)(a² + ab + b²), which can sometimes simplify the process. We'll explore both methods to give you a comprehensive understanding.

Method 1: Expanding the Cubes Directly

The first method we'll use involves expanding the cubic expressions directly. This means we'll multiply out (x+4)³ and (x-3)³ and then subtract the results. While this method can be a bit lengthy, it's straightforward and helps solidify our understanding of algebraic expansion. Let's start by expanding (x+4)³. This means we need to multiply (x+4) by itself three times: (x+4)(x+4)(x+4). We can start by multiplying the first two factors: (x+4)(x+4) = x² + 8x + 16. Now, we multiply this result by the remaining (x+4): (x² + 8x + 16)(x+4) = x³ + 12x² + 48x + 64. So, (x+4)³ = x³ + 12x² + 48x + 64. Next, we expand (x-3)³ in a similar way: (x-3)(x-3)(x-3). First, multiply the first two factors: (x-3)(x-3) = x² - 6x + 9. Then, multiply this result by the remaining (x-3): (x² - 6x + 9)(x-3) = x³ - 9x² + 27x - 27. So, (x-3)³ = x³ - 9x² + 27x - 27. Now that we have expanded both cubic expressions, we can substitute them back into the original equation: (x³ + 12x² + 48x + 64) - (x³ - 9x² + 27x - 27) = 343. Distribute the negative sign: x³ + 12x² + 48x + 64 - x³ + 9x² - 27x + 27 = 343. Combine like terms: 21x² + 21x + 91 = 343. Subtract 343 from both sides to set the equation to zero: 21x² + 21x - 252 = 0. Divide the entire equation by 21 to simplify: x² + x - 12 = 0. Now we have a quadratic equation that we can solve by factoring.

Solving the Quadratic Equation

We've simplified the original equation to a quadratic equation: x² + x - 12 = 0. Now, we need to solve for 'x'. There are a few ways to solve quadratic equations, including factoring, using the quadratic formula, or completing the square. In this case, factoring seems like the easiest approach. Factoring involves finding two numbers that multiply to -12 and add up to 1 (the coefficient of the 'x' term). Those numbers are 4 and -3. So, we can factor the quadratic equation as: (x + 4)(x - 3) = 0. Now, we can use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This means that either x + 4 = 0 or x - 3 = 0. Solving these two equations gives us: x = -4 or x = 3. So, we have two possible solutions for 'x': -4 and 3. It's always a good idea to check our solutions by plugging them back into the original equation to make sure they work.

Method 2: Using the a³ - b³ Identity

Now, let's explore a second method for solving the equation using the identity a³ - b³ = (a - b)(a² + ab + b²). This method can sometimes be more efficient, especially when dealing with complex expressions. In our case, we can let a = (x+4) and b = (x-3). So, our equation (x+4)³ - (x-3)³ = 343 can be rewritten as a³ - b³ = 343. Now, let's apply the identity: (a - b)(a² + ab + b²) = 343. First, we need to find (a - b): (x+4) - (x-3) = x + 4 - x + 3 = 7. So, (a - b) = 7. Now, let's find (a² + ab + b²). We already know that a = (x+4) and b = (x-3), so: a² = (x+4)² = x² + 8x + 16 b² = (x-3)² = x² - 6x + 9 ab = (x+4)(x-3) = x² + x - 12 Now, we can substitute these expressions into (a² + ab + b²): (x² + 8x + 16) + (x² + x - 12) + (x² - 6x + 9) = 3x² + 3x + 13. So, our equation becomes: 7(3x² + 3x + 13) = 343. Divide both sides by 7: 3x² + 3x + 13 = 49. Subtract 49 from both sides to set the equation to zero: 3x² + 3x - 36 = 0. Divide the entire equation by 3 to simplify: x² + x - 12 = 0. Notice that this is the same quadratic equation we obtained in Method 1! We already know how to solve this equation, and the solutions are x = -4 and x = 3.

Checking the Solutions

We found two possible solutions for 'x': x = -4 and x = 3. Now, let's check if these solutions actually work by plugging them back into the original equation. First, let's check x = -4: ((-4)+4)³ - ((-4)-3)³ = (0)³ - (-7)³ = 0 - (-343) = 343. So, x = -4 is a valid solution. Now, let's check x = 3: ((3)+4)³ - ((3)-3)³ = (7)³ - (0)³ = 343 - 0 = 343. So, x = 3 is also a valid solution. Both solutions work, so we have successfully solved the equation!

Final Answer

Alright, guys! We've successfully navigated this algebraic challenge. By expanding the cubic expressions and simplifying the equation, we found two solutions for 'x': x = -4 and x = 3. We also verified these solutions by plugging them back into the original equation. Remember, practice makes perfect, so keep tackling those math problems, and you'll become a pro in no time! Whether you prefer expanding the cubes directly or using the a³ - b³ identity, the key is to understand the steps and apply them carefully. Math can be fun, especially when you break down complex problems into manageable steps. Keep exploring, keep learning, and keep shining!

Conclusion

In this comprehensive guide, we've walked through a step-by-step solution to the equation (x+4)³ - (x-3)³ = 343. We explored two methods: expanding the cubes directly and using the a³ - b³ identity. Both methods led us to the same quadratic equation, which we solved by factoring. We then verified our solutions, confirming that x = -4 and x = 3 are indeed the correct answers. This problem highlights the importance of algebraic manipulation and the power of using identities to simplify complex expressions. By understanding these techniques, you can confidently tackle a wide range of algebraic challenges. Remember, math is a journey, and every problem you solve is a step forward. Keep practicing, keep exploring, and most importantly, keep enjoying the process of learning!