Solving (x^2+4)(x^2-6) A Step-by-Step Guide

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Hey guys! Ever stumbled upon an equation that looks like a jumble of numbers and letters? Don't worry, we've all been there. Today, we're going to break down a specific equation: (x2+4)(x2-6). This might seem intimidating at first, but trust me, with a little bit of algebraic magic, we'll solve it together. So, grab your thinking caps, and let's dive in!

Understanding the Basics

Before we jump into solving this equation, let's quickly brush up on some fundamental algebraic concepts. These concepts will serve as our toolkit as we navigate the equation. So, let's get started with understanding the basics.

  • Variables: Think of variables, like our 'x,' as placeholders for unknown numbers. Our goal is often to figure out what number 'x' represents.
  • Exponents: The little number perched up high, like the '2' in x^2, is an exponent. It tells us how many times to multiply the base (in this case, 'x') by itself. So, x^2 means x * x.
  • Parentheses: Parentheses are like VIP sections in an equation. We need to deal with what's inside them first.
  • The Distributive Property: This is our key to unlocking equations like this. It states that a(b + c) = ab + ac. Basically, we multiply the term outside the parentheses by each term inside.

Applying the Distributive Property

The distributive property is the backbone of expanding expressions like the one we're tackling. It's like carefully unpacking a box to see all its contents. In our case, we have two sets of parentheses multiplied together: (x^2 + 4)(x^2 - 6). To expand this, we'll treat the first set of parentheses as a single term and distribute it across the second set, and then distribute each term within the first set individually. Let's break it down step by step.

First, we take the first term in the first parenthesis, which is x^2, and multiply it by each term in the second parenthesis. This gives us x^2 * x^2 and x^2 * -6. Remember that when multiplying terms with exponents, you add the exponents if the bases are the same. So, x^2 * x^2 becomes x^(2+2) which simplifies to x^4. And x^2 * -6 simply becomes -6x^2.

Next, we take the second term in the first parenthesis, which is +4, and multiply it by each term in the second parenthesis. This gives us 4 * x^2 and 4 * -6. These simplify to 4x^2 and -24, respectively. Now we have all the pieces of our expanded expression.

Combining Like Terms

Once we've applied the distributive property, we often end up with an expression that has several terms. However, not all terms are created equal. Some terms are like peas in a pod – they have the same variable raised to the same power. These are called 'like terms,' and we can combine them to simplify our expression.

In our expanded expression, we have terms with x^4, x^2, and constant terms (numbers without variables). The x^4 term stands alone, as there are no other terms with x raised to the fourth power. However, we have two terms with x^2: -6x^2 and +4x^2. These are like terms, and we can combine them by simply adding their coefficients (the numbers in front of the variable). So, -6x^2 + 4x^2 becomes -2x^2. Similarly, the constant term -24 stands alone, as there are no other constant terms to combine it with.

Step-by-Step Solution

Okay, let's get down to business and solve this equation step by step. We'll use the distributive property and combine like terms to get our final answer. Make sure you're following along, guys, because this is where the magic happens!

  1. Apply the Distributive Property: As we discussed, we'll multiply each term in the first set of parentheses by each term in the second set. This gives us:

    (x^2 + 4)(x^2 - 6) = x^2 * x^2 + x^2 * (-6) + 4 * x^2 + 4 * (-6)

  2. Simplify: Now, let's simplify each of these multiplications:

    x^2 * x^2 = x^4 x^2 * (-6) = -6x^2 4 * x^2 = 4x^2 4 * (-6) = -24

So, our equation now looks like this:

x^4 - 6x^2 + 4x^2 - 24

  1. Combine Like Terms: Remember, like terms have the same variable raised to the same power. In this case, -6x^2 and 4x^2 are like terms. Let's combine them:

    -6x^2 + 4x^2 = -2x^2

Now, our equation is:

x^4 - 2x^2 - 24

And there you have it! We've successfully expanded and simplified the equation. The final result is x^4 - 2x^2 - 24.

Common Mistakes to Avoid

Alright, let's talk about some common pitfalls to watch out for when solving equations like this. We want to make sure we're not tripping over any algebraic hurdles, right? Here are a few mistakes you should definitely avoid.

Forgetting the Distributive Property

One of the biggest errors is not properly applying the distributive property. Remember, you need to multiply each term in the first set of parentheses by every term in the second set. If you miss even one multiplication, your final answer will be off. So, double-check that you've distributed correctly.

Sign Errors

Sign errors are sneaky little devils that can trip up even the most seasoned mathletes. Pay close attention to the signs (positive or negative) of each term, especially when multiplying. A negative times a negative is a positive, and a positive times a negative is a negative. Get those signs straight!

Combining Unlike Terms

Remember, you can only combine terms that are like terms – that is, they have the same variable raised to the same power. You can't combine x^4 with x^2, for example. It's like trying to add apples and oranges; they're just not the same. So, make sure you're only combining like terms.

Order of Operations

Ah, the infamous order of operations! It's a fundamental rule in math, and it's crucial for solving equations correctly. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? This tells you the order in which to perform operations. If you ignore the order of operations, you're likely to end up with the wrong answer.

Real-World Applications

Now, you might be thinking, "Okay, this is great, but when am I ever going to use this in real life?" Well, you'd be surprised! Algebraic equations like this pop up in various fields, from engineering to physics to computer science. Understanding how to solve them can open doors to some pretty cool stuff.

Engineering

In engineering, these equations can be used to model all sorts of systems, like the trajectory of a projectile or the stress on a structural beam. Engineers need to be able to solve these equations to design safe and efficient structures and machines.

Physics

Physics is full of equations that describe the world around us, and many of them involve polynomials like the one we've been working with. For example, equations of motion, which describe how objects move, often involve quadratic or higher-order polynomials.

Computer Science

In computer science, polynomials are used in algorithms for computer graphics, data compression, and cryptography. If you're interested in creating special effects for movies or developing secure ways to transmit data, understanding these equations is essential.

Conclusion

So, guys, we've successfully unraveled the equation (x2+4)(x2-6). We've broken it down step by step, from understanding the basics of algebra to applying the distributive property and combining like terms. We've also discussed common mistakes to avoid and explored some real-world applications of these types of equations.

Remember, practice makes perfect! The more you work with these concepts, the more comfortable you'll become. So, keep those thinking caps on, and keep exploring the wonderful world of algebra! You've got this!