Solving The Equation: X(x^2-x-6)/(x^2-4) = 0
Hey guys! Let's dive into solving this equation step-by-step. We've got a bit of a complex expression here, but don't worry, we'll break it down and make it super easy to understand. The equation we are tackling today is: x(x2-x-6)/(x2-4) = 0. Our main goal is to find all the values of x that make this equation true. This involves a little bit of algebra, factoring, and some careful consideration of potential pitfalls like division by zero. So, grab your thinking caps, and let's get started!
Step 1: Factoring the Quadratic Expressions
Okay, so the first thing we need to do when we see polynomial equations like this is to factor them. Factoring helps us simplify the equation and find the roots more easily. We have two quadratic expressions in our equation: x^2 - x - 6 in the numerator and x^2 - 4 in the denominator. Let’s tackle the numerator first.
Factoring x^2 - x - 6
To factor the quadratic expression x^2 - x - 6, we need to find two numbers that multiply to -6 and add up to -1 (the coefficient of the x term). Think about it for a second… What two numbers fit this description? If you guessed -3 and 2, you're spot on! Because -3 multiplied by 2 equals -6, and -3 plus 2 equals -1. So, we can rewrite the quadratic expression as follows:
x^2 - x - 6 = (x - 3)(x + 2)
See? Not too scary, right? Factoring is like breaking down a complex problem into smaller, more manageable pieces. Now, let's move on to the denominator.
Factoring x^2 - 4
The denominator, x^2 - 4, might look familiar to some of you. This is a classic example of a difference of squares. A difference of squares is an expression in the form a^2 - b^2, which can be factored as (a - b)(a + b). In our case, x^2 - 4 can be seen as x^2 - 2^2. So, applying the difference of squares factorization, we get:
x^2 - 4 = (x - 2)(x + 2)
Awesome! We've successfully factored both the numerator and the denominator. This is a crucial step because it allows us to simplify our equation and identify potential solutions. Trust me, it’s like having a secret weapon in our math-solving arsenal!
Step 2: Rewriting the Equation with Factored Expressions
Now that we've factored both the numerator and the denominator, let's rewrite the original equation with these factored expressions. This step makes the equation much clearer and sets us up perfectly for finding the solutions. Remember our original equation? It was:
x(x^2 - x - 6) / (x^2 - 4) = 0
We factored x^2 - x - 6 into (x - 3)(x + 2) and x^2 - 4 into (x - 2)(x + 2). So, let's substitute these back into the equation. This gives us:
x(x - 3)(x + 2) / (x - 2)(x + 2) = 0
See how much simpler it looks now? It’s like we’ve taken a complicated puzzle and started putting the pieces in the right place. The next step involves identifying any common factors we can cancel out, which will further simplify the equation.
Step 3: Identifying and Canceling Common Factors
Alright, time to put on our detective hats and spot some common factors! Looking at our rewritten equation, x(x - 3)(x + 2) / (x - 2)(x + 2) = 0, do you notice anything that appears in both the numerator and the denominator?
That's right! We've got (x + 2) in both the numerator and the denominator. This means we can cancel them out. But, and this is a big but, we need to be careful here. Canceling (x + 2) means we are essentially dividing both the numerator and the denominator by (x + 2). We can only do this if (x + 2) is not equal to zero. Why? Because dividing by zero is a major no-no in mathematics. It's like the mathematical equivalent of a black hole – things just break down!
So, before we go ahead and cancel, let’s make a note that x cannot be -2. If x were -2, then (x + 2) would be zero, and we’d be dividing by zero. We need to remember this restriction because it affects our final solution set.
Now that we've made that crucial observation, we can safely cancel out the (x + 2) terms. This simplifies our equation to:
x(x - 3) / (x - 2) = 0
Wow, that looks much cleaner! We’re making great progress. Now, let's move on to the next step, where we'll identify any other restrictions and then finally solve for x. Keep up the awesome work, guys!
Step 4: Identifying Restrictions and Solving for x
Okay, we've simplified our equation to x(x - 3) / (x - 2) = 0. Before we jump into solving for x, it's super important to identify any restrictions on x. Remember, we already found one restriction when we canceled out (x + 2): x cannot be -2. But let's take another look at our simplified equation. Do you see any other values of x that would cause problems?
Think about the denominator, (x - 2). What happens if x is 2? If x is 2, the denominator becomes zero, and again, we’re dividing by zero – which is a big no-no! So, we have another restriction: x cannot be 2.
Now that we’ve identified all our restrictions (x ≠-2 and x ≠2), we can focus on finding the values of x that make the equation true. For a fraction to equal zero, the numerator must be zero (while the denominator is not zero, which we've already taken care of with our restrictions). So, we need to find the values of x that make x(x - 3) equal to zero.
Setting the Numerator to Zero
We have x(x - 3) = 0. This equation is already factored, which is fantastic! To find the solutions, we simply set each factor equal to zero:
- x = 0
- x - 3 = 0
Solving the second equation, x - 3 = 0, we simply add 3 to both sides, which gives us x = 3.
So, we have two potential solutions: x = 0 and x = 3. These are the values of x that make the numerator zero. Now, let's make sure these solutions don't violate our restrictions. Remember, x cannot be -2 or 2. Our solutions are 0 and 3, which are perfectly fine! They don't conflict with our restrictions. We’re in the clear!
Step 5: Stating the Solution Set
Alright, we've reached the final step! We’ve done the hard work, factored the equation, identified the restrictions, and solved for x. Now, we just need to put it all together and state our solution set. Remember, the solution set is the set of all values of x that make the original equation true.
We found two solutions: x = 0 and x = 3. We also made sure that these solutions don't violate any restrictions. So, our solution set consists of these two values.
Therefore, the solution set for the equation x(x^2 - x - 6) / (x^2 - 4) = 0 is {0, 3}.
And that’s it, guys! We’ve successfully navigated through this equation, factoring, simplifying, and solving. Give yourselves a pat on the back – you’ve earned it! Math can be challenging, but breaking it down step by step makes it much more manageable. Plus, it’s super satisfying when you finally crack the code and find the solution. Great job!
Final Answer
The solution set of the equation x(x2-x-6)/(x2-4) = 0 is A. {0, 3}.