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

In this comprehensive guide, we will delve into the process of solving the quadratic equation $x^2 - 8x + 41 = 0$. Quadratic equations, which take the general form of $ax^2 + bx + c = 0$, are fundamental in mathematics and have wide-ranging applications in various fields such as physics, engineering, and economics. The solutions to these equations, also known as roots or zeros, can be found using a variety of methods, including factoring, completing the square, and the quadratic formula. In this particular case, we will employ the quadratic formula, a powerful tool that provides a direct solution for any quadratic equation, regardless of its factorability. Our main keyword here is to solve quadratic equations. Understanding how to solve quadratic equations is crucial for many mathematical and real-world problems.

The quadratic formula is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$. Identifying these coefficients correctly is the first crucial step in applying the quadratic formula. In our equation, $x^2 - 8x + 41 = 0$, we can easily identify the coefficients as follows:

• $a = 1$ (the coefficient of $x^2$)
• $b = -8$ (the coefficient of $x$)
• $c = 41$ (the constant term)

Now that we have identified the coefficients, we can substitute them into the quadratic formula. This substitution requires careful attention to detail to ensure that the signs and values are correctly placed within the formula. The next step involves simplifying the expression, particularly the discriminant, which is the term under the square root. The discriminant plays a vital role in determining the nature of the roots – whether they are real or complex. This is a key aspect of solving quadratic equations.

Applying the Quadratic Formula

Having identified the coefficients $a = 1$, $b = -8$, and $c = 41$ from the equation $x^2 - 8x + 41 = 0$, we now substitute these values into the quadratic formula:

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

The first step in simplifying the expression is to evaluate the terms inside the square root and the denominator. We have $-(-8) = 8$ in the numerator and $2(1) = 2$ in the denominator. Inside the square root, we calculate $(-8)^2 = 64$ and $4(1)(41) = 164$. Thus, the expression becomes:

$x = \frac{8 \pm \sqrt{64 - 164}}{2}$

Next, we simplify the expression under the square root: $64 - 164 = -100$. This gives us:

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

The presence of a negative number under the square root indicates that the solutions will be complex numbers. This is because the square root of a negative number is not a real number. To proceed, we recall that the imaginary unit, denoted by $i$, is defined as $i = \sqrt{-1}$. Therefore, we can rewrite $\sqrt{-100}$ as $\sqrt{100 \cdot -1} = \sqrt{100} \cdot \sqrt{-1} = 10i$. Our main keyword is complex numbers, as understanding them is crucial for solving quadratic equations with negative discriminants.

Now, the expression becomes:

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

Finally, we divide both the real and imaginary parts of the numerator by the denominator, 2:

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

$x = 4 \pm 5i$

Thus, the solutions to the quadratic equation $x^2 - 8x + 41 = 0$ are $x = 4 + 5i$ and $x = 4 - 5i$. These are complex conjugate roots, which is a common occurrence when solving quadratic equations with a negative discriminant. The process of solving quadratic equations often involves simplifying complex numbers, and this step is a critical part of the solution.

Understanding the Discriminant

The discriminant, denoted as $b^2 - 4ac$, is a crucial component of the quadratic formula that provides valuable information about the nature of the roots of a quadratic equation. It is the expression under the square root in the quadratic formula, and its value determines whether the roots are real and distinct, real and equal, or complex conjugates. The discriminant is a key concept when solving quadratic equations.

1. If $b^2 - 4ac > 0$: The equation has two distinct real roots. This means that the parabola represented by the quadratic equation intersects the x-axis at two different points.
2. If $b^2 - 4ac = 0$: The equation has one real root (or two equal real roots). In this case, the parabola touches the x-axis at exactly one point.
3. If $b^2 - 4ac < 0$: The equation has two complex conjugate roots. This means that the parabola does not intersect the x-axis. Our main keyword here is the nature of roots, as understanding this is critical for solving quadratic equations.

In our example, $x^2 - 8x + 41 = 0$, the discriminant is:

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

Since the discriminant is $-100$, which is less than 0, the equation has two complex conjugate roots, as we found earlier. Understanding the discriminant allows us to predict the type of solutions we will obtain even before fully applying the quadratic formula. This is an important aspect of solving quadratic equations efficiently.

Alternative Methods for Solving Quadratic Equations

While the quadratic formula is a universally applicable method for solving quadratic equations, there are other techniques that can be used depending on the specific form of the equation. These methods include factoring and completing the square. Factoring is often the quickest method when it is applicable, but it is not always straightforward, especially when the roots are not rational. Completing the square, on the other hand, is a more general method that can be used to solve any quadratic equation, and it also provides a way to derive the quadratic formula itself.

Factoring

Factoring involves expressing the quadratic equation in the form $(px + q)(rx + s) = 0$, where $p$, $q$, $r$, and $s$ are constants. If we can factor the quadratic expression, we can then set each factor equal to zero and solve for $x$. However, factoring is not always easy, especially when the coefficients are large or the roots are irrational or complex. In the case of $x^2 - 8x + 41 = 0$, factoring is not a viable option because there are no two integers that multiply to 41 and add up to -8. This highlights the importance of having other methods, like the quadratic formula, at our disposal. Our main keyword here is factoring, as it is a fundamental technique for solving quadratic equations when applicable.

Completing the Square

Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved. The general idea is to manipulate the equation $ax^2 + bx + c = 0$ into the form $(x + h)^2 = k$, where $h$ and $k$ are constants. This method is particularly useful for understanding the structure of quadratic equations and for deriving the quadratic formula. Our main keyword here is completing the square, as it is a powerful method for solving quadratic equations and for understanding their properties.

To complete the square for the equation $x^2 - 8x + 41 = 0$, we first subtract 41 from both sides:

$x^2 - 8x = -41$

Next, we take half of the coefficient of the $x$ term (-8), which is -4, and square it: $(-4)^2 = 16$. We add this value to both sides of the equation:

$x^2 - 8x + 16 = -41 + 16$

Now, the left side of the equation is a perfect square trinomial, which can be factored as $(x - 4)^2$. The right side simplifies to -25:

$(x - 4)^2 = -25$

Taking the square root of both sides, we get:

$x - 4 = \pm \sqrt{-25}$

$x - 4 = \pm 5i$

Finally, we add 4 to both sides to solve for $x$:

$x = 4 \pm 5i$

As we can see, completing the square yields the same solutions as the quadratic formula, $x = 4 + 5i$ and $x = 4 - 5i$. This method provides an alternative approach to solving quadratic equations and reinforces the concept of complex roots when the discriminant is negative.

Conclusion

In conclusion, we have successfully solved the quadratic equation $x^2 - 8x + 41 = 0$ using the quadratic formula and the method of completing the square. The solutions are the complex conjugates $x = 4 + 5i$ and $x = 4 - 5i$. We also discussed the importance of the discriminant in determining the nature of the roots and explored alternative methods for solving quadratic equations, such as factoring and completing the square. Understanding these methods and concepts is essential for mastering quadratic equations and their applications in various mathematical and real-world contexts. The ability to solve quadratic equations is a fundamental skill in mathematics, and a thorough understanding of the different techniques available ensures that you can tackle any quadratic equation effectively.