Solving (x+3)^2+(x+3)-2=0: A Step-by-Step Guide
Hey there, math enthusiasts! Today, we're diving into solving the equation (x+3)^2 + (x+3) - 2 = 0. Don't worry, it's not as scary as it looks! We'll break it down step-by-step so you can easily follow along. So, grab your pencils and let's get started!
Understanding the Equation
Before we jump into the solution, let's take a moment to understand what we're dealing with. The equation (x+3)^2 + (x+3) - 2 = 0 is a quadratic equation in disguise. Notice how the expression (x+3) appears twice? This is a clue that we can use a clever substitution to simplify the equation and make it easier to solve. Our main goal here is to find the values of x that make this equation true. In other words, we're looking for the roots or solutions of the equation. This is a common type of problem in algebra, and mastering it will definitely help you in your mathematical journey. Remember, practice makes perfect, so don't hesitate to try out similar problems to build your confidence and understanding.
Step 1: Substitution
To make things simpler, we'll use a technique called substitution. This involves replacing a more complex expression with a single variable. In this case, let's substitute u = x + 3. This means wherever we see (x + 3) in the equation, we'll replace it with u. This seemingly small step can make a huge difference in simplifying the equation. By making this substitution, we transform the original equation into a much more manageable form. The main goal is to reduce the complexity and make it easier to apply standard solving techniques. This is a common strategy in mathematics – break down a complex problem into smaller, simpler steps. Now, let's see how this substitution actually changes the equation.
Substituting u into our equation, (x+3)^2 + (x+3) - 2 = 0, gives us:
u^2 + u - 2 = 0
See how much cleaner that looks? We've transformed a potentially intimidating equation into a standard quadratic equation that we can easily solve. This substitution is a crucial step in our process. It allows us to apply familiar techniques without getting bogged down in the original complexity. Always look for opportunities to simplify equations through substitution, it's a powerful tool in your mathematical arsenal. Now that we have our simplified equation, let's move on to the next step: factoring.
Step 2: Factoring the Quadratic Equation
Now that we have the equation u^2 + u - 2 = 0, we can solve it by factoring. Factoring involves breaking down the quadratic expression into the product of two binomials. Think of it as reversing the process of expanding brackets. We're looking for two numbers that multiply to give us -2 (the constant term) and add up to give us 1 (the coefficient of the u term). This might sound tricky at first, but with a little practice, you'll get the hang of it. Factoring is a fundamental skill in algebra, and it's essential for solving many types of equations. It's like having a key that unlocks the solution. So, let's find those two special numbers!
After some thought, we can see that the numbers 2 and -1 satisfy these conditions (2 * -1 = -2 and 2 + (-1) = 1). Therefore, we can factor the equation as:
(u + 2)(u - 1) = 0
Great job! We've successfully factored the quadratic equation. Now we have two factors that, when multiplied together, equal zero. This is a crucial point because it leads us to the next step: finding the values of u that make each factor equal to zero. Remember, if the product of two things is zero, at least one of them must be zero. This principle will allow us to find the possible solutions for u. Let's move on and see how this works.
Step 3: Solving for u
Now that we have the factored equation (u + 2)(u - 1) = 0, we can use the zero-product property to find the solutions for u. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, this means either (u + 2) = 0 or (u - 1) = 0. This gives us two simple linear equations to solve. Solving for u in each of these equations will give us the possible values for u. This step is like unlocking the hidden values of u that make our equation true. It's a direct application of a fundamental mathematical principle, and it brings us closer to our final solution. So, let's solve these equations and see what values we get for u.
Solving these equations, we get:
- u + 2 = 0 => u = -2
- u - 1 = 0 => u = 1
So, we have two possible values for u: -2 and 1. But remember, we're not trying to solve for u; we're trying to solve for x. We used the substitution u = x + 3 to simplify the equation, and now we need to reverse that substitution to find the values of x. This is like retracing our steps to get back to our original goal. We've come a long way, but the final piece of the puzzle is just around the corner. Let's move on to the next step and see how we can find the values of x.
Step 4: Substituting Back to Find x
We've found the values of u, but remember, we're trying to solve for x. So, we need to substitute back our original expression u = x + 3. This means we'll replace u with (x + 3) in our solutions. We have two values for u: -2 and 1. So, we'll set up two separate equations, one for each value of u. This step is crucial because it connects our simplified solution back to the original variable we were interested in. It's like translating our findings back into the language of the original problem. Now, let's see how this substitution helps us find the values of x. This is the final stretch, guys, we're almost there!
Substituting back, we get:
- x + 3 = -2
- x + 3 = 1
Now we have two simple equations that we can easily solve for x. This is the moment of truth where we finally find the solutions to our original equation. Each equation represents a possible value for x that makes the entire equation true. Solving these equations is like the final click in a combination lock, revealing the hidden answer. Let's go ahead and solve for x in each case.
Step 5: Solving for x
Now, let's solve the two equations we got in the previous step:
- x + 3 = -2
To solve for x, we subtract 3 from both sides:
x = -2 - 3 x = -5
- x + 3 = 1
Similarly, subtract 3 from both sides:
x = 1 - 3 x = -2
So, we have two solutions for x: -5 and -2. These are the values that, when plugged back into the original equation (x+3)^2 + (x+3) - 2 = 0, will make the equation true. We've successfully navigated through the problem, using substitution, factoring, and the zero-product property to arrive at our solutions. This is a great accomplishment, guys! But, as good mathematicians, we should always verify our solutions to make sure they are correct. Let's move on to the final step and check our answers.
Step 6: Verifying the Solutions
To make sure our solutions are correct, we need to plug them back into the original equation (x+3)^2 + (x+3) - 2 = 0 and see if they satisfy the equation. This is a crucial step in any problem-solving process, not just in mathematics. It's like a final check to ensure that everything is correct and that we haven't made any mistakes along the way. Verifying our solutions gives us confidence in our answer and helps us avoid careless errors. So, let's take our solutions, x = -5 and x = -2, and plug them back into the original equation to see if they work.
Let's check x = -5:
((-5) + 3)^2 + ((-5) + 3) - 2 = (-2)^2 + (-2) - 2 = 4 - 2 - 2 = 0
So, x = -5 is a solution.
Now, let's check x = -2:
((-2) + 3)^2 + ((-2) + 3) - 2 = (1)^2 + (1) - 2 = 1 + 1 - 2 = 0
So, x = -2 is also a solution.
Conclusion
Therefore, the solutions to the equation (x+3)^2 + (x+3) - 2 = 0 are x = -5 and x = -2. Awesome job, guys! We successfully solved the equation by using substitution, factoring, the zero-product property, and verifying our solutions. This is a great example of how we can break down complex problems into smaller, more manageable steps. Remember, mathematics is not about memorizing formulas; it's about understanding the concepts and applying them creatively. Keep practicing, keep exploring, and keep having fun with math! You've got this!