Solving Quadratic Equations: A Step-by-Step Guide

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Hey math enthusiasts! Today, we're diving into the world of quadratic equations and figuring out how to crack the code to find their solutions. Specifically, we're going to solve the equation x2−8x+97=0x^2 - 8x + 97 = 0. Don't worry if it seems intimidating at first; we'll break it down step by step to make sure everyone understands. Let's get started!

Understanding Quadratic Equations

First off, what exactly is a quadratic equation? Well, it's an equation that can be written in the general form of ax2+bx+c=0ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. These equations are super important in various fields, from physics and engineering to finance and even video game development! The solutions to a quadratic equation, often called roots or zeros, are the values of x that make the equation true. These roots can be real numbers, but they can also be complex numbers, which involve the imaginary unit, often denoted as i, where i=−1i = \sqrt{-1}. Our equation, x2−8x+97=0x^2 - 8x + 97 = 0, fits this format perfectly. Here, a = 1, b = -8, and c = 97. We’re dealing with a quadratic equation, which means we're going to find two solutions, even if they're complex! The ability to solve these equations is a fundamental skill in algebra and is essential for tackling more complex mathematical problems down the road. Keep in mind that quadratic equations represent parabolas when graphed, and the solutions are the points where the parabola intersects the x-axis (the x-intercepts). When the parabola doesn't cross the x-axis, the solutions are complex numbers. This understanding of quadratic equations is the cornerstone of many mathematical concepts. Being able to solve them allows us to explore a wide range of real-world problems. For example, knowing how to solve a quadratic equation will help calculate the trajectory of a projectile or determine the optimal pricing strategy for a product. Moreover, the techniques we use to solve these equations are often applied in more advanced mathematics, such as calculus and linear algebra. So, understanding these concepts can really boost your overall problem-solving skills and enhance your academic performance.

The Quadratic Formula: Your Secret Weapon

Alright, so how do we actually solve this equation? We're going to use the quadratic formula. It's a lifesaver, honestly! The quadratic formula is a universal method that works for any quadratic equation. It's given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where a, b, and c are the coefficients from our quadratic equation. In our case, x2−8x+97=0x^2 - 8x + 97 = 0, we have a = 1, b = -8, and c = 97. Now, let’s plug those values into the formula. We'll start by substituting the values of a, b, and c into the quadratic formula. It's really just a matter of careful substitution. Remember that the formula includes the plus or minus sign (±), which means we will end up with two solutions, one using addition and one using subtraction. This is why quadratic equations can have up to two solutions. After substituting, you'll want to carefully simplify the equation step by step. This may involve squaring, multiplying, and finding the square root of a number. Make sure to keep track of your calculations. One common mistake is getting the order of operations wrong (PEMDAS/BODMAS). So, make sure you know what to do before trying to solve the problem. The discriminant, part of the formula under the square root, tells us about the type of solutions we have. If the discriminant is positive, we get two real solutions. If it’s zero, we get one real solution. If it's negative, we get two complex solutions, which is the case in this problem.

Let’s go ahead and substitute the values: $x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(97)}}{2(1)}$ See? Not too bad, right? We’ll keep simplifying this step by step.

Calculation Steps and Solution

Okay, let's break down the calculations. We have: $x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(97)}}{2(1)}$

  • First, simplify the terms inside the square root (the discriminant): (−8)2=64(-8)^2 = 64 and 4∗1∗97=3884 * 1 * 97 = 388. So, we get: 64−388=−32464 - 388 = -324
  • Now our equation looks like this: $x = \frac{8 \pm \sqrt{-324}}{2}$
  • Here's where the imaginary numbers come into play! The square root of -324 is 18i18i, because i=−1i = \sqrt{-1}. So, −324=324∗−1=18i\sqrt{-324} = \sqrt{324} * \sqrt{-1} = 18i
  • Our equation is now: $x = \frac{8 \pm 18i}{2}$
  • Finally, divide both the real and imaginary parts by 2: $x = 4 \pm 9i$

Therefore, the two solutions are x=4+9ix = 4 + 9i and x=4−9ix = 4 - 9i. This means that the correct answer is option C. Congratulations! You've successfully solved a quadratic equation with complex roots! Remember that when the discriminant (the value under the square root) is negative, the solutions will always be complex numbers, involving the imaginary unit i. It also means the graph of the quadratic equation (a parabola) will not intersect the x-axis. This process can be applied to all quadratic equations and is a fundamental skill in algebra.

Why This Matters and Real-World Applications

Solving quadratic equations isn't just an academic exercise; it has real-world applications. For example, it is used in physics, to determine the trajectory of a projectile, like a ball thrown in the air or a rocket launched into space. It's used in engineering, to design bridges, buildings, and other structures that can withstand various forces. Understanding quadratic equations is also used in economics and finance. They can be used to model market behaviors, calculate profit maximization, and analyze investment returns. Additionally, they are used in computer graphics and game development, in order to create realistic animations and simulate movements. So, knowing how to solve these equations equips you with the tools to solve complex problems and model various phenomena. The ability to use the quadratic formula enables you to understand and calculate a wide range of problems.

Tips for Success

  • Practice, practice, practice! The more you solve quadratic equations, the more comfortable you’ll become with the process. Try working through different examples and varying the coefficients to improve your skills.
  • Master the quadratic formula! Memorize it! You'll be using it a lot. Knowing the quadratic formula is a key to success. Practicing it can also help you solve problems more easily. The more you know the formula, the faster you will be able to solve the equation.
  • Pay attention to signs! Be extra careful with positive and negative signs, especially when substituting values into the quadratic formula. Double-check your calculations! A simple sign error can change the entire solution.
  • Understand the discriminant. Know what the discriminant tells you about the nature of the solutions (real, complex, and how many). Understanding the discriminant will help you anticipate the type of answer you’ll get.
  • Break it down. Don't rush through the steps. Take your time, and break down the problem into smaller, manageable parts. Taking your time will help ensure that your calculations are correct.
  • Check your work. If you have time, plug your solutions back into the original equation to make sure they're correct. This is a great way to confirm your answer. You can use this method to check your work and find the mistakes easily.

Conclusion

And there you have it, folks! We've successfully solved our quadratic equation and explored the beauty of the quadratic formula. Remember, math can be challenging, but with the right approach and practice, anyone can master it. Keep exploring, keep learning, and keep solving! You've got this!

I hope this guide has been helpful! If you have any questions, feel free to ask. Happy calculating!