Solving Quadratic Equations: Finding The Solutions

Hey everyone! Today, we're diving into the world of quadratic equations and figuring out how to find their solutions. Specifically, we'll tackle the equation $x^2 = 16x - 65$. Don't worry, it's not as scary as it looks! We'll break it down step-by-step, making sure you understand every move. Our goal is to find the values of x that make this equation true. We're also going to explore how complex numbers can pop up as solutions, which is pretty cool. Get ready to flex those math muscles and learn something new!

Understanding Quadratic Equations

Alright, before we jump into the problem, let's make sure we're all on the same page about what a quadratic equation even is. Simply put, it's an equation that can be written in the general form: $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero. The highest power of the variable (in this case, x) is 2, which is why it's called a quadratic equation (think quad as in square). These equations are super important in algebra because they pop up everywhere – from physics and engineering to finance and even in computer graphics! The solutions to a quadratic equation are the values of x that satisfy the equation. Sometimes, we'll get two real number solutions, sometimes one (when the solutions are repeated), and sometimes, as we'll see today, we get complex number solutions. Understanding quadratic equations is like having a superpower in the math world; it allows you to solve a huge variety of problems. So, when we talk about solutions, we are finding the roots of the equation, the x-values that make the equation balance perfectly, essentially the x-intercepts when you graph the quadratic function (a parabola). These solutions can be found using different methods, such as factoring, completing the square, or the quadratic formula, which is a key part of our problem today.

Rearranging the Equation

Now, let's get back to our specific equation: $x^2 = 16x - 65$. The first thing we want to do is get everything on one side of the equation and set it equal to zero, which is the standard form of a quadratic equation. This way, we can easily identify our a, b, and c values. So, we'll subtract $16x$ and add $65$ to both sides. Doing this gives us: $x^2 - 16x + 65 = 0$. See, now it looks much more familiar. We've got our x squared term, our x term, and our constant term all nicely arranged. This is also known as the standard form. When an equation is written this way, it's easier to use a variety of solving techniques such as completing the square, factoring, or the quadratic formula. Each of these methods will take us to the same solution, but we might choose one over the other depending on the specific equation and our own preferences. With our equation in the form $ax^2 + bx + c = 0$, we have a=1, b=-16 and c=65. This will prove useful later as we begin to solve for x.

Choosing a Solution Method

There are several ways to solve a quadratic equation, including factoring, completing the square, and using the quadratic formula. Given our equation, we could try factoring, but it might not be immediately obvious if there are two numbers that multiply to 65 and add up to -16. Completing the square is another option, and it's a useful technique, but it can sometimes involve a few more steps. In this case, we'll go with the quadratic formula. The quadratic formula is a universal tool and will always work for any quadratic equation. It's especially handy when factoring seems difficult or when we suspect we might end up with complex solutions. It's a lifesaver, really! The quadratic formula is: $x = rac{-b

sqrt{b^2 - 4ac}}{2a}$. This formula will give us the solutions for x directly, and we can just plug in our values of a, b, and c that we found earlier. So, let’s get started!

Applying the Quadratic Formula

Alright, let's plug the values from our equation, $x^2 - 16x + 65 = 0$, into the quadratic formula. Remember, we have a = 1, b = -16, and c = 65. So, the formula becomes: $x = rac{-(-16)

sqrt{(-16)^2 - 4 * 1 * 65}}{2 * 1}$. Simplifying step by step is key here so let's start with the double negative. The negative of a negative turns positive so now we have $x = rac{16

sqrt{(-16)^2 - 4 * 1 * 65}}{2 * 1}$. Next, we'll compute inside the square root. We'll square -16 to get 256. Then we multiply 4 * 1 * 65 = 260. So now we get $x = rac{16

sqrt{256 - 260}}{2 * 1}$. Continuing on, we'll simplify and get $x = rac{16

sqrt{-4}}{2}$. Now, here's where things get interesting! We see a negative number inside the square root. What does this mean? It means we're going to get complex number solutions. Remember that the square root of -1 is defined as i – the imaginary unit. This is how complex numbers enter the scene! We'll proceed to simplify this expression, knowing that the square root of -4 is 2i. So the equation becomes $x = rac{16

• 2i}{2}$.

Simplifying for the Solutions

Okay, we're almost there! Now, we have $x = rac{16

• 2i}{2}$. The next step is to divide both the real and imaginary parts by 2 to get our final solutions. So, we'll divide both 16 and -2i by 2. This gives us $x = 8
• i$. That's one solution! Since the quadratic formula gives us two possible solutions, let's remember that the original formula had a plus or minus sign (±) in front of the square root. This means we actually have two solutions: one with a plus sign and one with a minus sign. So, our two solutions are: $x = 8 + i$ and $x = 8 - i$. Both of these are complex numbers. Remember when we said our solutions would be in the form $x = 8 + bi$ and $x = 8 - bi$? Well, now we've found them! This shows how solving quadratic equations can lead to solutions that are not just real numbers, but also complex numbers. Complex numbers have a real part and an imaginary part, and they're essential in various fields of mathematics, physics, and engineering. The solutions are often expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. We see the solutions are $x = 8 + i$ and $x = 8 - i$, or in the requested form, where b=1.

Conclusion: The Solutions Revealed

So there you have it, guys! We've successfully solved the quadratic equation $x^2 = 16x - 65$ and found its solutions. We started with the equation, rearranged it, and used the quadratic formula. We discovered that the solutions are complex numbers, $x = 8 + i$ and $x = 8 - i$, which are in the form $x = 8 + bi$ and $x = 8 - bi$ where b=1. This journey highlighted the importance of understanding different types of numbers and the power of the quadratic formula. The process shows that solving quadratic equations can involve complex numbers, and it's a critical tool in your algebra toolbox. Keep practicing, and you'll become a pro at solving these types of equations in no time! Keep in mind that understanding these concepts is not just about getting the right answer; it's about developing your problem-solving skills and your understanding of how mathematics works. Congratulations on solving the quadratic equation!

I hope you found this helpful. If you have any questions or want to try some more examples, feel free to ask in the comments. Happy solving!