Solving X In The Equation X² - 8x + 41 = 0 A Comprehensive Guide

Hey guys! Let's dive into solving a quadratic equation today. We've got a fun one: x² - 8x + 41 = 0. Now, when we look at this equation, we might try to factor it, but it quickly becomes clear that simple factoring isn't going to work here. So, what's our next move? That's right, we're going to use the quadratic formula! This formula is a trusty tool for solving any quadratic equation in the form ax² + bx + c = 0. In our case, a is 1, b is -8, and c is 41. The quadratic formula looks like this:

The Quadratic Formula

The quadratic formula is given by:

x = [-b ± √(b² - 4ac)] / 2a

It might look a bit intimidating at first, but trust me, it's easier than it seems once you break it down. The key here is to carefully substitute the values of a, b, and c from our equation into the formula. So, let's do that now. We have a = 1, b = -8, and c = 41. Plugging these values into the quadratic formula gives us:

x = [-(-8) ± √((-8)² - 4 * 1 * 41)] / (2 * 1)

Alright, we've made our substitutions. Now, the next step is to simplify this expression. First, let's take care of the negative signs and the multiplication. We have:

x = [8 ± √(64 - 164)] / 2

Notice that we have 64 minus 164 under the square root. This is going to give us a negative number, which means we'll be dealing with imaginary numbers – exciting stuff! Let's simplify further:

x = [8 ± √(-100)] / 2

Now we have the square root of -100. Remember that the square root of a negative number involves the imaginary unit i, where i is defined as the square root of -1. So, the square root of -100 can be written as:

√(-100) = √(100 * -1) = √100 * √(-1) = 10i

Great! Now we can substitute this back into our equation:

x = [8 ± 10i] / 2

We're almost there! Our final step is to divide both terms in the numerator by 2:

x = 4 ± 5i

And there we have it! Our solutions for x are 4 + 5i and 4 - 5i. So, if we look back at the options provided, we can see that the correct answer is D. x = 4 ± 5i. This means the roots of the quadratic equation are complex numbers, which is totally fine and happens quite often in math. Remember, the quadratic formula is your friend when you can't factor a quadratic equation easily. Keep practicing, and you'll become a pro at solving these in no time! This quadratic formula is a cornerstone in algebra and is frequently used in various mathematical and engineering contexts. Mastering it ensures you can tackle a wide array of problems effectively. Don't hesitate to revisit and practice this method until it becomes second nature.

Understanding Complex Roots

When solving quadratic equations, it’s essential to understand that the roots can be real or complex. In our example, we encountered a situation where the discriminant (the part under the square root, b² - 4ac) was negative. Specifically, we had √(-100), which led us to complex roots. The discriminant provides crucial information about the nature of the roots. If b² - 4ac is positive, we have two distinct real roots. If it’s zero, we have one real root (or two identical real roots). And, as we saw, if it’s negative, we have two complex conjugate roots. Complex roots always come in conjugate pairs, meaning if a + bi is a root, then a - bi is also a root. This is a fundamental property of quadratic equations with real coefficients. Recognizing this pattern can help you quickly verify your solutions. For instance, in our problem, we found the roots 4 + 5i and 4 - 5i, which are indeed complex conjugates. Understanding the nature of roots not only helps in solving equations but also provides deeper insights into the behavior of quadratic functions and their graphs. Remember, complex roots mean that the parabola represented by the quadratic equation does not intersect the x-axis. This is a crucial concept in graphical analysis and helps in visualizing the solutions of quadratic equations. So, next time you encounter a negative discriminant, you'll know you're in the realm of complex numbers, and that’s perfectly okay! Embrace the complexity; it's what makes math so fascinating!

Alternative Methods and Verifications

While the quadratic formula is a reliable method, it's always beneficial to explore alternative approaches and verification techniques. One such method is completing the square. Completing the square involves transforming the quadratic equation into a perfect square trinomial, which can then be easily solved. Let’s briefly look at how this could be applied to our equation, x² - 8x + 41 = 0. First, we rearrange the equation to isolate the x terms: x² - 8x = -41. Next, we complete the square by adding the square of half the coefficient of x to both sides. Half the coefficient of x is -8/2 = -4, and squaring it gives us (-4)² = 16. So, we add 16 to both sides: x² - 8x + 16 = -41 + 16. This simplifies to (x - 4)² = -25. Taking the square root of both sides, we get x - 4 = ±√(-25), which leads to x - 4 = ±5i. Finally, solving for x, we get x = 4 ± 5i, the same solution we obtained using the quadratic formula. This demonstrates the versatility of different methods and how they can be used to verify results. Another helpful technique is to use Vieta's formulas, which relate the coefficients of a polynomial to the sums and products of its roots. For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a. In our case, the sum of the roots should be -(-8)/1 = 8, and the product should be 41/1 = 41. Let’s verify this with our solutions, 4 + 5i and 4 - 5i. The sum is (4 + 5i) + (4 - 5i) = 8, and the product is (4 + 5i)(4 - 5i) = 16 - (5i)² = 16 - (-25) = 41. Both conditions are satisfied, providing further confidence in our solutions. By exploring these alternative methods and verification techniques, you not only reinforce your understanding but also develop a more robust problem-solving toolkit. So, keep exploring, keep questioning, and keep verifying!

Conclusion

In conclusion, solving the quadratic equation x² - 8x + 41 = 0 led us to the complex solutions x = 4 ± 5i. We utilized the quadratic formula, a powerful tool for tackling any quadratic equation, and explored the significance of a negative discriminant, which indicates complex roots. Additionally, we briefly touched upon alternative methods like completing the square and verification techniques such as Vieta's formulas. Remember, math is not just about finding the correct answer; it's about understanding the underlying concepts and developing a versatile problem-solving approach. Keep practicing, exploring different methods, and always verifying your solutions. You've got this, guys! Happy solving! Understanding the nuances of quadratic equations extends beyond just finding solutions; it lays a foundation for more advanced mathematical concepts. The ability to recognize and handle complex roots is crucial in fields like electrical engineering, where imaginary numbers are used to describe alternating current circuits, and in quantum mechanics, where complex numbers are fundamental to the wave functions that describe the behavior of particles. Furthermore, the techniques we discussed, such as completing the square and using Vieta's formulas, are not limited to quadratic equations. They serve as stepping stones for understanding higher-degree polynomials and more complex algebraic structures. The journey through solving quadratic equations is a microcosm of the broader mathematical landscape, where each solved problem opens doors to new challenges and discoveries. So, continue to embrace the process, cultivate your problem-solving skills, and let the beauty of mathematics unfold before you. The more you engage with these concepts, the more you'll appreciate the elegance and power of mathematical thinking. Keep pushing your boundaries, and you'll be amazed at what you can achieve!