Solving Quadratic Equations Finding M For One Real Solution

Hey everyone! Today, we're diving into the fascinating world of quadratic equations, those tricky expressions with the squared term that can sometimes seem like they're speaking a different language. But fear not, because we're going to break down a specific problem step-by-step, making sure you understand not just the how, but also the why behind each move. So, let's get started!

The Challenge: One Real Solution

Our mission, should we choose to accept it (and we do!), involves the equation $-2x^2 + 32x - k = 0$. This is a classic quadratic equation, where $k$ is just a constant, a number that stays put. The real twist comes when we're told that this equation has exactly one real solution when $k = m + 1$. This is our key to unlocking the mystery. To understand what this means, we need to recall the concept of discriminant.

The Discriminant: Your Quadratic Equation Decoder Ring

The discriminant is a magical part of the quadratic formula that tells us how many real solutions a quadratic equation has. Remember the quadratic formula? It's that beastly expression:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Well, the discriminant is the part under the square root: $b^2 - 4ac$. This little expression holds the key to our solution. Let's break down why:

• If $b^2 - 4ac > 0$, the equation has two distinct real solutions. Think of it like this: the square root gives you a positive number, which you add and subtract from $-b$, giving you two different answers for $x$.
• If $b^2 - 4ac = 0$, the equation has exactly one real solution. The square root is zero, so you're just left with $-b/2a$, one single answer for $x$.
• If $b^2 - 4ac < 0$, the equation has no real solutions. The square root of a negative number is not a real number, so we venture into the realm of imaginary numbers.

In our case, we want exactly one real solution, so we need the discriminant to be equal to zero. This is the crucial concept that will guide us through the rest of the problem. Remember, a quadratic equation having one real solution means its discriminant is zero. This is the cornerstone of solving this type of problem, and it is essential to understand the relationship between the discriminant and the number of real solutions to solve similar quadratic equations. Make sure that you are very clear on the different conditions of the discriminant and how they affect the nature of the roots of the quadratic equation. We will apply this knowledge in the upcoming steps to solve for the value of m. This foundational understanding is not just for this particular problem but also vital for tackling various problems in algebra and calculus that involve quadratic equations. Let's dive deeper into how the discriminant helps us in determining the nature of the roots, as this understanding is fundamental. When the discriminant is positive, the square root part of the quadratic formula yields two real values, leading to two distinct roots. When it is negative, we encounter the square root of a negative number, resulting in complex roots, which are not real. Only when the discriminant is zero do we have a single real root, as the square root part vanishes, leaving us with just one solution. So, remember the significance of the discriminant! It's more than just a formula; it's a gateway to understanding the behavior and solutions of quadratic equations. Grasping this concept is not just about memorizing a rule but about developing a deeper insight into how mathematical tools can reveal the properties of equations. We will now use this understanding to solve the given problem by setting up the equation with the discriminant equal to zero and solving for the unknown variable, m. This practical application will further solidify your understanding of the concept and demonstrate how abstract theory translates into concrete solutions. Keep this key idea in mind as we move forward, and you'll find that complex problems like this become much more manageable. Understanding the discriminant will empower you to tackle similar challenges with confidence and precision. So, let's put this powerful knowledge into action!

Applying the Discriminant to Our Problem

Now, let's bring it back to our specific equation: $-2x^2 + 32x - k = 0$. We can identify our coefficients: $a = -2$, $b = 32$, and $c = -k$. Remember, $k$ is going to be replaced with $m + 1$ soon, but for now, let's keep it as $k$.

Our discriminant, $b^2 - 4ac$, becomes:

$32^2 - 4(-2)(-k)$

Simplifying this, we get:

$1024 - 8k$

Since we want exactly one real solution, we set the discriminant to zero:

$1024 - 8k = 0$

Now we solve for $k$:

$8k = 1024$

$k = 128$

So, we've found that for our equation to have one real solution, $k$ must be 128. But remember, the problem asks for the value of $m$, and we know that $k = m + 1$. Let's not forget this crucial step in solving the problem! It's easy to get caught up in the calculations and forget what the question is ultimately asking. Always double-check your work and ensure you're answering the correct question. In our case, we've found the value of $k$, but the problem specifically asks for $m$. This highlights the importance of reading the question carefully and understanding what is being asked. Missing this step can lead to an incorrect final answer, even if all the previous calculations are correct. Think of it like a treasure hunt: you've found the treasure chest, but you still need to open it to get the gold inside! Finding $k$ is like finding the chest, but solving for $m$ is like getting the gold. We're almost there, so let's not lose focus now. The relationship between $k$ and $m$ is the final piece of the puzzle, and we're about to put it in place. Remember, it's not just about the mathematical process; it's about understanding the problem's context and objectives. This attention to detail is what separates a good problem-solver from a great one. So, keep your eye on the prize, and let's find that value of $m$! This part of the process is just as important as the earlier calculations, as it completes the solution and ensures we're giving the correct answer. We're in the home stretch now, so let's make sure we cross the finish line with precision and clarity. Now that we have the value of $k$, the final step is within our grasp. We'll use the equation $k = m + 1$ to find $m$, and then we'll have successfully solved the problem! So, let's make the final connection and reveal the value of $m$. It's a satisfying moment when all the pieces come together, and we're just about to experience that. Keep the momentum going, and let's wrap this up with a flourish!

The Final Step: Solving for m

Since $k = m + 1$, we can substitute $k = 128$:

$128 = m + 1$

Subtracting 1 from both sides, we get:

$m = 127$

And there you have it! The value of $m$ is 127. We've successfully navigated through the quadratic equation, understood the discriminant, and solved for our unknown. Remember, the key here was understanding that exactly one real solution meant the discriminant had to be zero. This understanding of the discriminant is key to solving this problem! It's not just about plugging numbers into a formula; it's about grasping the concept behind the mathematics. We started with the quadratic equation and the condition for one real solution, and we systematically worked our way to the answer. This process highlights the importance of breaking down complex problems into smaller, manageable steps. Each step, from identifying the coefficients to setting the discriminant to zero and solving for $k$, was crucial. Then, remembering the relationship $k = m + 1$, we were able to find the value of $m$. This step-by-step approach is a valuable skill in mathematics and problem-solving in general. By methodically addressing each part of the problem, we avoided confusion and were able to arrive at the correct solution. Think of it like building a house: each brick and each stage of construction is essential to the final structure. Similarly, each step in solving a mathematical problem contributes to the overall solution. This structured approach not only helps in solving the current problem but also builds confidence and proficiency for future challenges. So, remember the power of breaking down complexity into manageable pieces. Whether it's a quadratic equation or a real-world problem, this strategy will serve you well. We've successfully solved for $m$ by utilizing our understanding of the discriminant and applying a systematic approach. This combination of conceptual knowledge and methodical problem-solving is what leads to success in mathematics. Now that we've found the value of $m$, we can reflect on the entire process and appreciate how each element contributed to the final answer. So, let's take a moment to recap the key steps and insights we've gained from this exercise. We've not only solved a specific problem but also reinforced fundamental mathematical principles that will be invaluable in tackling other challenges. This is the real reward of learning mathematics: not just the answer to a specific question, but the acquisition of skills and understanding that can be applied in a wide range of contexts.

Conclusion: Quadratic Equations Unlocked!

So, guys, we've tackled a tricky quadratic equation and emerged victorious! Remember, the key takeaways are: understand the discriminant, read the question carefully, and break down complex problems into smaller steps. Keep practicing, and you'll be quadratic equation masters in no time!