Is (x+2) Cubed Equal To 2x(x Squared Minus 1) An Algebraic Exploration

Hey everyone! Today, we're tackling a fascinating algebraic puzzle that's sure to flex your mathematical muscles. We're going to explore the equation (x+2)³ = 2x(x² - 1) and meticulously check if it holds true. So, grab your thinking caps, and let's embark on this mathematical adventure together!

Decoding the Cube: Expanding (x+2)³

Okay, first things first, let's unravel the mystery of (x+2)³. This means we're essentially multiplying (x+2) by itself three times: (x+2) * (x+2) * (x+2). To make things easier, we'll break it down step-by-step. Let's start by expanding the first two (x+2) terms. Remember the distributive property, guys? It's our best friend here! We'll multiply each term in the first parenthesis by each term in the second.

So, (x+2) * (x+2) becomes x * x + x * 2 + 2 * x + 2 * 2. Simplifying this, we get x² + 2x + 2x + 4. Combining those like terms, we arrive at x² + 4x + 4. Awesome! We've conquered the first part. But hold on, the journey isn't over yet. We still need to multiply this result by another (x+2).

Now, we're faced with (x² + 4x + 4) * (x+2). Don't fret! We'll use the distributive property again, but this time it's a bit more extensive. We'll multiply each term in the first parenthesis (x² + 4x + 4) by both x and 2 in the second parenthesis. Buckle up, here we go!

Let's start by multiplying each term by x: x² * x + 4x * x + 4 * x. This gives us x³ + 4x² + 4x. Great! Now, let's multiply each term by 2: x² * 2 + 4x * 2 + 4 * 2. This results in 2x² + 8x + 8. We're almost there, guys! Now, we need to combine all the terms we've gathered.

We have x³ + 4x² + 4x + 2x² + 8x + 8. Time to hunt for those like terms and bring them together! We have one x³ term, so that stays as is. For the x² terms, we have 4x² and 2x², which combine to give us 6x². Moving on to the x terms, we have 4x and 8x, which add up to 12x. And finally, we have the constant term, 8. Putting it all together, we have x³ + 6x² + 12x + 8. Woohoo! We've successfully expanded (x+2)³! That was quite a journey, but we made it. Remember, practice makes perfect, so don't be afraid to tackle similar expansions. You've got this!

Unveiling the Right Side: Simplifying 2x(x² - 1)

Alright, mathletes, now that we've conquered the left side of our equation, let's turn our attention to the right side: 2x(x² - 1). This part looks a bit more straightforward, but we still need to be meticulous in our simplification. Our key tool here, once again, is the distributive property. It's like the Swiss Army knife of algebra!

So, we need to distribute that 2x across the terms inside the parenthesis (x² - 1). This means we'll multiply 2x by x² and then multiply 2x by -1. Let's break it down step-by-step to keep things crystal clear.

First up, 2x multiplied by x². Remember your exponent rules, guys! When multiplying terms with the same base, we add the exponents. So, x is really x to the power of 1 (x¹). Thus, x¹ * x² becomes x^(1+2) which simplifies to x³. And don't forget the coefficient! 2 * 1 (the implicit coefficient in front of x²) is 2. So, 2x * x² gives us 2x³.

Now, let's move on to the second part: 2x multiplied by -1. This is pretty straightforward: 2x * -1 is simply -2x. Excellent! We've tackled both parts of the distribution.

Putting it all together, 2x(x² - 1) simplifies to 2x³ - 2x. See? That wasn't so bad! We've successfully simplified the right side of our equation. We're making great progress in our quest to determine if the equation holds true. Just one more step to go – the grand comparison!

The Grand Showdown: Comparing Both Sides and Solving for x

Okay, math detectives, the moment of truth has arrived! We've diligently simplified both sides of our equation, and now it's time for the grand showdown. We'll compare our simplified expressions and see if they're equal. If they are, then the equation holds true for all values of x. If not, we'll need to figure out for which values of x the equation is satisfied. This is where the real algebraic magic happens!

So, let's recap. We found that (x+2)³ expands to x³ + 6x² + 12x + 8. And we simplified 2x(x² - 1) to 2x³ - 2x. Now, we set them equal to each other: x³ + 6x² + 12x + 8 = 2x³ - 2x. The stage is set! Let the solving commence!

Our goal now is to isolate x, which means getting all the x terms on one side of the equation and the constants on the other. To do this, we'll start by subtracting x³ from both sides. This gives us 6x² + 12x + 8 = x³ - 2x. Next, let's subtract 6x² from both sides: 12x + 8 = x³ - 6x² - 2x. Then, we subtract 12x from both sides: 8 = x³ - 6x² - 14x. Finally, subtract 8 from both sides to set the equation to zero: 0 = x³ - 6x² - 14x - 8.

Now, guys, we're facing a cubic equation. Solving cubic equations can be a bit tricky, but don't worry, we have tools at our disposal! One common method is to try and factor the cubic expression. Factoring involves finding expressions that, when multiplied together, give us our original cubic expression. This might involve trying different values for x to see if they make the expression equal to zero. These values are called roots or solutions of the equation.

Alternatively, we can use numerical methods or graphing tools to approximate the solutions. Graphing the cubic equation can give us a visual representation of the roots, where the graph intersects the x-axis. Numerical methods, like the Newton-Raphson method, can provide accurate approximations of the roots.

In this particular case, finding the roots might involve some trial and error or the use of more advanced techniques. However, the key takeaway here is the process: we expanded, simplified, set the expressions equal, and now we're ready to solve for x. This is the heart of algebra, guys! We've taken a complex problem and broken it down into manageable steps. Keep practicing, and you'll become master equation solvers in no time!

The Verdict: Does the Equation Hold True?

Alright, everyone, we've reached the climax of our mathematical investigation! We've expanded, simplified, and compared both sides of the equation (x+2)³ = 2x(x² - 1). Now, it's time to deliver the verdict: does this equation hold true? Drumroll, please...

As we saw in the previous section, after expanding and simplifying, we arrived at the cubic equation 0 = x³ - 6x² - 14x - 8. This is a crucial point. If the original equation held true for all values of x, then this cubic equation would have to be an identity, meaning it would be true for every possible value of x. However, a cubic equation has at most three solutions (roots). This means it's not an identity.

Therefore, the original equation (x+2)³ = 2x(x² - 1) does not hold true for all values of x. It's a conditional equation, meaning it's only true for specific values of x – namely, the roots of the cubic equation we derived.

So, the answer to our initial question is a resounding no. The equation is not universally true. It's a fascinating result, guys! It highlights the importance of careful algebraic manipulation and the fact that not all equations are created equal. Some hold true universally, while others are only true under specific conditions.

To find the specific values of x that do satisfy the equation, we would need to solve the cubic equation 0 = x³ - 6x² - 14x - 8. As we discussed earlier, this can be done through factoring (if possible), numerical methods, or graphing. These methods would reveal the three specific values of x for which the equation (x+2)³ = 2x(x² - 1) is indeed true.

But for now, we've successfully answered the core question: the equation is not a universal truth. We've flexed our algebraic muscles, explored the depths of equation solving, and emerged victorious with a clear and definitive answer. Give yourselves a pat on the back, guys! You've conquered another mathematical challenge!

Wrapping Up: Key Takeaways and Further Exploration

We've reached the end of our algebraic expedition, guys, but the journey of mathematical learning never truly ends! Let's take a moment to recap the key takeaways from our exploration of the equation (x+2)³ = 2x(x² - 1), and then I'll suggest some avenues for further mathematical adventures.

• Expansion is key: We saw how expanding expressions like (x+2)³ is a fundamental skill in algebra. Mastering techniques like the distributive property is crucial for simplifying and solving equations.
• Simplification is your friend: Simplifying both sides of an equation before comparing them is a golden rule. It makes the equation easier to handle and reduces the risk of errors.
• Not all equations are equal: We discovered that not all equations hold true for all values of the variable. Some are conditional, meaning they're only true for specific solutions.
• Cubic equations can be tricky: Solving cubic equations can be a challenge, but we learned about the tools at our disposal, such as factoring, numerical methods, and graphing.
• The power of comparison: Setting simplified expressions equal to each other allows us to solve for the unknown variable and determine the truth of the equation.

So, where can you go from here, math enthusiasts? Here are a few ideas:

• Practice more expansions: Try expanding other expressions like (x-3)³, (2x+1)³, or even higher powers. The more you practice, the more comfortable you'll become.
• Tackle more cubic equations: Explore different methods for solving cubic equations, such as the rational root theorem or Cardano's method. This will deepen your understanding of polynomial equations.
• Investigate graphing tools: Use graphing calculators or online tools to visualize equations and their solutions. This can provide valuable insights into the behavior of functions.
• Dive into numerical methods: Learn about numerical techniques like the Newton-Raphson method for approximating solutions to equations. This is a powerful tool in many areas of mathematics and science.
• Explore other types of equations: Challenge yourself with quadratic equations, radical equations, or even trigonometric equations. The world of equations is vast and fascinating!

Remember, guys, mathematics is like a muscle – the more you use it, the stronger it gets. So, keep practicing, keep exploring, and never stop asking questions. The world of mathematics is full of wonders waiting to be discovered! Thanks for joining me on this algebraic adventure. Until next time, keep those math brains buzzing!