Solving $x^2 - 8x + 41 = 0$: A Step-by-Step Guide

Hey guys! Let's dive into solving a quadratic equation today. We're going to tackle the equation $x^2 - 8x + 41 = 0$. Quadratic equations can seem daunting, but with the right approach, they're totally manageable. So, grab your calculators, and let’s get started!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

Where a, b, and c are constants, and x is the variable we want to solve for. In our case, the equation is $x^2 - 8x + 41 = 0$. Here, a = 1, b = -8, and c = 41. Understanding these coefficients is crucial for applying the quadratic formula, which we’ll use to find the solutions for x.

Solving quadratic equations is a fundamental skill in algebra, and it pops up in various areas of math and science. From physics problems involving projectile motion to engineering designs requiring parabolic curves, knowing how to solve these equations is super useful. Plus, it's a great way to sharpen your problem-solving skills. There are a few methods to solve quadratic equations, such as factoring, completing the square, and using the quadratic formula. Factoring is great when it's straightforward, but it’s not always easy to spot the factors. Completing the square is another method that’s always reliable but can get a bit messy with the numbers. That’s why we often turn to the quadratic formula – it’s a one-size-fits-all solution that works every time. So, let’s dive into how to use it for our equation.

The Quadratic Formula: Our Go-To Solution

The quadratic formula is a powerful tool that gives us the solutions for any quadratic equation. It’s derived from the method of completing the square and looks like this:

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

This formula might look a bit intimidating, but don't worry! We'll break it down step by step. The plus-minus symbol ($\pm$) indicates that there are usually two solutions: one where we add the square root and one where we subtract it. These solutions are also known as the roots of the equation. To apply the quadratic formula, we need to identify the values of a, b, and c from our equation. Remember, in our equation $x^2 - 8x + 41 = 0$, we have a = 1, b = -8, and c = 41. Now, let's plug these values into the formula and see what we get.

Using the quadratic formula is like having a universal key that unlocks the solutions to any quadratic equation. It’s super handy because it doesn’t rely on guesswork or tricky factoring techniques. Whether the solutions are real numbers, complex numbers, or irrational numbers, the quadratic formula handles them all. This makes it an indispensable part of your mathematical toolkit. So, next time you're faced with a quadratic equation, remember this trusty formula. It’s your best friend in these situations. Let’s go ahead and substitute our values into the formula and watch the magic happen!

Step-by-Step Solution

Let's plug the values a = 1, b = -8, and c = 41 into the quadratic formula:

$x = \frac{-(-8) e \sqrt{(-8)^2 - 4(1)(41)}}{2(1)}$

First, we simplify the expression inside the square root:

$(-8)^2 - 4(1)(41) = 64 - 164 = -100$

Now, we substitute this back into the formula:

$x = \frac{8 e \sqrt{-100}}{2}$

Since we have a negative number inside the square root, we know that the solutions will be complex numbers. Remember that $\sqrt{-1} = i$, where i is the imaginary unit. So, $\sqrt{-100} = \sqrt{100} imes \sqrt{-1} = 10i$. Now we can rewrite our equation as:

$x = \frac{8 e 10i}{2}$

Next, we divide both terms in the numerator by 2:

$x = 4 e 5i$

So, the solutions are $x = 4 + 5i$ and $x = 4 - 5i$.

Each step in the solution process is important. Breaking down the quadratic formula into manageable parts makes it easier to understand and apply. From squaring b to handling the negative inside the square root, each operation builds toward the final solution. And remember, when you encounter a negative number under the square root, that's your cue to bring in the imaginary unit i. This is what allows us to express the solutions as complex numbers. So, let’s review what we’ve done and confirm our answer.

Verifying the Solutions

Our solutions are $x = 4 + 5i$ and $x = 4 - 5i$. To verify these solutions, we can plug them back into the original equation $x^2 - 8x + 41 = 0$ and see if they satisfy the equation. Let's start with $x = 4 + 5i$:

$(4 + 5i)^2 - 8(4 + 5i) + 41$

Expand $(4 + 5i)^2$:

$(4 + 5i)(4 + 5i) = 16 + 20i + 20i + 25i^2 = 16 + 40i - 25 = -9 + 40i$

Now, substitute back into the equation:

$-9 + 40i - 8(4 + 5i) + 41 = -9 + 40i - 32 - 40i + 41$

Combine like terms:

$(-9 - 32 + 41) + (40i - 40i) = 0$

So, $x = 4 + 5i$ is indeed a solution. Now let's check $x = 4 - 5i$:

$(4 - 5i)^2 - 8(4 - 5i) + 41$

Expand $(4 - 5i)^2$:

$(4 - 5i)(4 - 5i) = 16 - 20i - 20i + 25i^2 = 16 - 40i - 25 = -9 - 40i$

Substitute back into the equation:

$-9 - 40i - 8(4 - 5i) + 41 = -9 - 40i - 32 + 40i + 41$

Combine like terms:

$(-9 - 32 + 41) + (-40i + 40i) = 0$

So, $x = 4 - 5i$ is also a solution. Both solutions satisfy the original equation, so we can be confident in our answer.

Verifying your solutions is a critical step in problem-solving. It’s like the final seal of approval on your work. By plugging the solutions back into the original equation, you ensure that they hold true. This is especially important when dealing with complex numbers, as the calculations can sometimes get a bit tricky. Taking the time to verify your answers not only confirms your work but also reinforces your understanding of the concepts involved. So, always remember to double-check your solutions whenever possible.

Conclusion

We successfully solved the quadratic equation $x^2 - 8x + 41 = 0$ using the quadratic formula. Our solutions are $x = 4 + 5i$ and $x = 4 - 5i$. Remember, the quadratic formula is your best friend when dealing with quadratic equations, especially when factoring isn't straightforward. Keep practicing, and you'll become a pro at solving these equations in no time! Great job, guys! You nailed it!

The solutions are D. $x = 4 e 5i$