Solve Quadratic Equations: U-Substitution Method

Hey guys! Today, we're diving into the exciting world of quadratic equations and learning a neat trick called u-substitution to make solving them a breeze. We'll tackle the equation $(x+2)^2 - 2(x+2) - 15 = 0$ together, step by step, so you'll be a pro in no time. So, grab your pencils and let's get started!

Understanding the Power of U-Substitution

Before we jump into the specifics of this equation, let's chat about why u-substitution is such a handy tool in our mathematical arsenal. Think of it as a way to simplify complex expressions by replacing a chunk of them with a single, more manageable variable – our trusty friend, u. This technique is especially useful when we spot a repeating expression within an equation, as it allows us to transform a seemingly complicated problem into a more familiar form, like a standard quadratic equation.

In this case, we notice that the expression (x+2) appears twice in our equation: $(x+2)^2 - 2(x+2) - 15 = 0$. This is our cue to use u-substitution! By letting u represent (x+2), we can rewrite the entire equation in terms of u, making it look much cleaner and easier to solve. This substitution allows us to temporarily set aside the complexity of the original expression and focus on solving a simpler quadratic equation. Once we find the values of u, we can then easily substitute back to find the corresponding values of x. It's like a mathematical magic trick that makes challenging problems disappear, leaving behind a simpler path to the solution.

The beauty of u-substitution lies in its ability to transform a complex-looking equation into a manageable one, without altering the fundamental mathematical relationships. This is crucial because it ensures that the solutions we find for the simplified equation can be directly related back to the original problem. In essence, we're creating a temporary detour in our solving process, a detour that leads us to a clearer and more straightforward path to the final answer. So, with u-substitution in our toolbox, we're well-equipped to tackle a wide range of quadratic equations and other mathematical challenges.

Applying U-Substitution to Our Equation

Okay, let's get down to business and apply this u-substitution magic to our equation: $(x+2)^2 - 2(x+2) - 15 = 0$. Remember, the key is to identify the repeating expression, which in this case is, as we already discussed, (x+2). This is the expression we'll replace with our new variable, u. So, let's make it official:

Let u = (x+2)

Now, we can rewrite our equation by substituting u wherever we see (x+2). This gives us:

$u^2 - 2u - 15 = 0$

See how much simpler that looks? We've transformed a potentially intimidating equation into a standard quadratic equation that we can solve using familiar techniques. This is the power of u-substitution in action – it streamlines the problem and makes it much less daunting.

By making this strategic substitution, we've effectively isolated the quadratic structure of the equation, making it easier to apply standard solving methods. We've essentially peeled away the outer layers of complexity and revealed the core equation that we need to solve. This simplification not only makes the equation more visually appealing but also allows our brains to focus on the essential mathematical relationships at play. This is why u-substitution is such a valuable tool: it helps us declutter the problem and focus on the heart of the matter.

Now that we have our simplified equation in terms of u, we're ready to move on to the next step: solving for u. This will involve factoring the quadratic equation or using other techniques, such as the quadratic formula, to find the values of u that satisfy the equation. Once we have those values, we'll then substitute back to find the corresponding values of x. So, stay tuned as we continue our journey toward solving this quadratic equation using the magic of u-substitution!

Solving the Quadratic Equation in Terms of U

Now that we've successfully substituted u = (x+2) and transformed our equation into $u^2 - 2u - 15 = 0$, it's time to solve for u. We have a classic quadratic equation staring back at us, and there are a couple of ways we can tackle it. One common method is factoring, and that's the route we'll take here because it's often the quickest way to find the solutions. Factoring involves breaking down the quadratic expression into the product of two binomials.

To factor $u^2 - 2u - 15$, we need to find two numbers that multiply to -15 and add up to -2. Think about the factors of -15: we have pairs like (1, -15), (-1, 15), (3, -5), and (-3, 5). Out of these pairs, the combination that adds up to -2 is 3 and -5. So, we can factor the quadratic expression as follows:

$(u + 3)(u - 5) = 0$

This factored form tells us that the product of (u + 3) and (u - 5) is zero. For this to be true, at least one of the factors must be zero. This leads us to two possible equations:

• u + 3 = 0 or
• u - 5 = 0

Solving each of these equations for u is a piece of cake. Subtracting 3 from both sides of the first equation gives us:

• u = -3

And adding 5 to both sides of the second equation gives us:

• u = 5

So, we've found our solutions for u: u = -3 and u = 5. But remember, we're not quite done yet! We've solved for u, but we ultimately want to find the values of x that satisfy the original equation. This means we need to substitute back and replace u with (x+2), which is what we'll do in the next step. We're on the home stretch now, guys!

Substituting Back to Solve for X

Alright, we've successfully solved for u, finding that u = -3 and u = 5. Now comes the crucial step of substituting back to find the values of x. Remember our initial substitution: u = (x + 2). We'll use this to replace u in our solutions and solve for x. This is where everything comes together, connecting our simplified equation back to the original problem.

Let's start with the first solution, u = -3. Substituting this into our u = (x + 2) equation, we get:

-3 = x + 2

To solve for x, we simply subtract 2 from both sides of the equation:

x = -3 - 2

x = -5

So, our first solution for x is -5. Now, let's tackle the second solution, u = 5. Substituting this into our u = (x + 2) equation, we get:

5 = x + 2

Again, we solve for x by subtracting 2 from both sides:

x = 5 - 2

x = 3

And there you have it! Our second solution for x is 3. We've successfully found both values of x that satisfy the original equation: x = -5 and x = 3. This demonstrates the power and elegance of u-substitution, allowing us to transform a seemingly complex problem into a series of simpler steps. We're now ready to confidently state our final answer, knowing that we've navigated the equation with precision and skill.

The Final Answer and Conclusion

After all our hard work and careful steps, we've arrived at the solution! We found that the solutions to the equation $(x+2)^2 - 2(x+2) - 15 = 0$ are x = -5 and x = 3. Looking back at the options provided, this corresponds to:

B. x = -5 and x = 3

So, B is the correct answer! We've successfully navigated this quadratic equation using the u-substitution technique, and hopefully, you've gained a clearer understanding of how this method works.

To recap, u-substitution is a powerful tool that allows us to simplify complex equations by replacing a repeating expression with a single variable, u. This transforms the equation into a more manageable form, often a standard quadratic equation that we can solve using factoring or other techniques. Once we find the solutions for u, we substitute back to find the corresponding values of x. This technique is not only efficient but also helps us to better understand the structure of the equation and the relationships between its components.

Remember, the key to mastering mathematics is practice. So, try applying u-substitution to other quadratic equations and similar problems. The more you practice, the more comfortable you'll become with the technique, and the more confident you'll feel tackling challenging equations. Keep up the great work, guys, and happy problem-solving!