Solving Quadratic Equations 5x² = -30x - 65

Hey guys! Let's dive into solving a quadratic equation today. We've got a fun one here: 5x² = -30x - 65. Quadratic equations can seem intimidating at first, but don't worry, we'll break it down step by step so it's super clear. Our goal is to find the values of x that make this equation true. We'll explore how to rearrange the equation into the standard quadratic form, discuss the quadratic formula (our trusty tool for solving these equations), and then apply it to find our solutions. So, buckle up, and let's get started!

Understanding Quadratic Equations

First things first, let's make sure we're all on the same page about what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. What does that mean? Well, it means the highest power of the variable (in our case, x) is 2. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero (otherwise, it wouldn't be quadratic anymore!). Think of a as the coefficient of the x² term, b as the coefficient of the x term, and c as the constant term. This form is super important because it sets us up perfectly for using the quadratic formula. Now, in our equation, 5x² = -30x - 65, we don't see that nice standard form right away. That's okay! Our first step is going to be rearranging the terms to get it into the ax² + bx + c = 0 format. This involves moving all the terms to one side of the equation, leaving zero on the other side. It's like organizing our ingredients before we start baking – it makes the whole process smoother. So, we'll add 30x and 65 to both sides of the equation. This keeps the equation balanced and gets us closer to where we need to be. This rearrangement is a crucial step because it allows us to easily identify the values of a, b, and c, which are essential for the next stage: using the quadratic formula. Without this standard form, applying the formula becomes much trickier, and we might end up with the wrong solutions. Think of it as setting the stage for the main event – the quadratic formula – which will ultimately reveal the values of x that satisfy our equation.

Transforming the Equation into Standard Form

Okay, let's get our hands dirty and transform the given equation into the standard quadratic form. We start with 5x² = -30x - 65. Remember, our goal is to get everything on one side and have zero on the other. To do this, we'll add 30x to both sides. This cancels out the -30x on the right side and introduces a new term on the left. We then add 65 to both sides. This cancels out the -65 on the right side and adds a constant term to the left. Doing so, we get 5x² + 30x + 65 = 0. Now we're talking! This looks much more like the ax² + bx + c = 0 form we were aiming for. From here, it's easy to spot the coefficients: a is 5 (the coefficient of x²), b is 30 (the coefficient of x), and c is 65 (the constant term). Identifying these values correctly is absolutely crucial. They are the key ingredients we need to plug into the quadratic formula. A small mistake here can throw off the entire calculation and lead to incorrect solutions. So, double-check these values before moving on! Now that we've successfully transformed the equation and identified a, b, and c, we're perfectly set up to use the quadratic formula. It's like having all the pieces of a puzzle – now we just need to put them together. The next step is where the magic really happens, and we'll see how this powerful formula helps us unlock the solutions for x.

Applying the Quadratic Formula

Alright, folks, here comes the star of the show: the quadratic formula! This formula is our best friend when it comes to solving quadratic equations. It's a general solution that works for any equation in the form ax² + bx + c = 0. The formula itself looks a bit intimidating at first, but trust me, it's not as scary as it seems. It states that the solutions for x are given by: x = (-b ± √(b² - 4ac)) / (2a). See those a, b, and c? That's why we made sure to identify them so carefully in the previous step! Now, let's plug in the values we found from our equation, 5x² + 30x + 65 = 0. We have a = 5, b = 30, and c = 65. Substituting these values into the formula, we get: x = (-30 ± √(30² - 4 * 5 * 65)) / (2 * 5). This might look like a mouthful, but we'll simplify it step by step. First, let's focus on the part under the square root, which is called the discriminant: b² - 4ac. This part is super important because it tells us about the nature of the solutions (whether they are real or complex). Calculating the discriminant, we have 30² - 4 * 5 * 65 = 900 - 1300 = -400. Uh oh! We've got a negative number under the square root. What does this mean? It means our solutions are going to be complex numbers, involving the imaginary unit i (where i² = -1). Don't worry, this is perfectly normal, and we'll handle it like pros. Now, let's continue simplifying the formula. We have x = (-30 ± √(-400)) / 10. The square root of -400 can be written as √(400 * -1) = √400 * √-1 = 20i. So, our equation becomes x = (-30 ± 20i) / 10. We're almost there! Now we just need to simplify this expression to get our final solutions.

Simplifying to Find the Solutions

Okay, we're in the home stretch now! We've got x = (-30 ± 20i) / 10. The final step is to simplify this expression to get our solutions for x. Notice that both -30 and 20 are divisible by 10, so we can simplify the fraction. We can divide both the numerator and the denominator by 10. This gives us x = (-3 ± 2i). And there we have it! We've found our solutions. Remember the ± sign? That means we actually have two solutions: one with the plus sign and one with the minus sign. So, our solutions are: x = -3 + 2i and x = -3 - 2i. These are complex conjugate pairs, which is a common occurrence when solving quadratic equations with a negative discriminant. These solutions tell us the values of x that make our original equation, 5x² = -30x - 65, true. They might not be the typical real number solutions we're used to, but they are perfectly valid solutions in the complex number system. Now, let's take a step back and think about what we've accomplished. We started with a quadratic equation, rearranged it into standard form, applied the quadratic formula, and simplified the result to find our complex solutions. That's a lot! You've successfully navigated the world of quadratic equations and complex numbers. Give yourself a pat on the back!

The Solutions

So, after all that hard work, let's clearly state our solutions. We found that the solutions to the equation 5x² = -30x - 65 are:

• x = -3 + 2i
• x = -3 - 2i

These are complex solutions, as we predicted when we encountered the negative discriminant. This means that there are no real number solutions to this equation. Graphically, this would mean that the parabola represented by the equation y = 5x² + 30x + 65 does not intersect the x-axis. Understanding the nature of the solutions (real or complex) is a key part of solving quadratic equations. The discriminant, b² - 4ac, is our helpful guide in this process. If it's positive, we have two distinct real solutions; if it's zero, we have one real solution (a repeated root); and if it's negative, as in our case, we have two complex conjugate solutions. Now that we've solved this equation, you can apply the same steps to tackle other quadratic equations. Remember the key steps: rearrange into standard form, identify a, b, and c, apply the quadratic formula, and simplify. With practice, you'll become a quadratic equation solving master! And remember, even if the solutions are complex, don't be intimidated – they are just as valid and important as real solutions in many mathematical contexts. So, keep exploring, keep practicing, and keep solving!

Conclusion

Fantastic job, everyone! We've successfully solved the quadratic equation 5x² = -30x - 65 and found the solutions x = -3 + 2i and x = -3 - 2i. We walked through the entire process, from rearranging the equation into standard form to applying the quadratic formula and simplifying the complex solutions. You've gained valuable experience in handling quadratic equations, even those with complex solutions. Remember, the quadratic formula is a powerful tool in your mathematical arsenal. It allows you to solve any quadratic equation, no matter how complicated it might seem at first. The key is to break the problem down into smaller, manageable steps. First, get the equation into the standard form ax² + bx + c = 0. Then, identify the coefficients a, b, and c. Next, carefully plug these values into the quadratic formula. Don't forget to pay close attention to the discriminant, b² - 4ac, as it tells you about the nature of the solutions. Finally, simplify the expression to find the solutions. Whether the solutions are real or complex, the process remains the same. Complex solutions might seem a bit mysterious at first, but they are an important part of mathematics and have applications in various fields, such as electrical engineering and quantum mechanics. So, embrace the complex numbers, and don't be afraid to work with them! Keep practicing, and you'll become more and more confident in your ability to solve quadratic equations. Now you're well-equipped to tackle more challenging problems and explore the fascinating world of mathematics even further. Keep up the great work!