Solving $x^2 + 164 = 16x$: Find Complex Solutions

In this article, we'll dive into solving the quadratic equation $x^2 + 164 = 16x$. Our main goal is to find the solutions and express them in the complex number form $x = a + bi$ and $x = a - bi$, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit ($\sqrt{-1}$). Quadratic equations, a cornerstone of algebra, appear frequently in various mathematical and real-world contexts. Understanding how to solve them, especially when the solutions involve complex numbers, is a crucial skill in mathematics.

Understanding Quadratic Equations and Complex Solutions

Before we jump into solving, let's briefly recap what quadratic equations are and what it means to have complex solutions. A quadratic equation is a polynomial equation of the second degree, generally written in the form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'x' is the variable. The solutions to a quadratic equation are also known as its roots.

When we attempt to solve a quadratic equation, we're essentially looking for the values of 'x' that make the equation true. These solutions can be real numbers, complex numbers, or a combination of both. Complex numbers arise when the discriminant ($b^2 - 4ac$) of the quadratic formula is negative. This indicates that the solutions involve the imaginary unit 'i', defined as $\sqrt{-1}$. Understanding complex numbers is crucial, especially when dealing with equations that don't have real-number solutions. The solutions we're aiming for are in the form $a + bi$ and $a - bi$, known as complex conjugates. These forms are essential in various fields, including electrical engineering, quantum mechanics, and advanced calculus.

Step-by-Step Solution to $x^2 + 164 = 16x$

Now, let’s tackle the equation $x^2 + 164 = 16x$ step by step. Our primary goal is to rearrange the equation into the standard quadratic form, apply the quadratic formula, and express the solutions in the required complex form. This process involves algebraic manipulation, careful calculation, and a clear understanding of complex number arithmetic.

1. Rearranging the Equation

The first step is to rewrite the equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. This involves moving all terms to one side of the equation, leaving zero on the other side. Starting with $x^2 + 164 = 16x$, we subtract $16x$ from both sides to get:

$x^2 - 16x + 164 = 0$

Now the equation is in the standard form, where we can identify the coefficients: a = 1, b = -16, and c = 164. Identifying these coefficients correctly is crucial for the next step, which involves applying the quadratic formula.

2. Applying the Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of any quadratic equation. It states that for an equation $ax^2 + bx + c = 0$, the solutions for 'x' are given by:

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

We can now substitute the values of a, b, and c that we identified earlier into the formula. This will give us the solutions to our equation. Substituting a = 1, b = -16, and c = 164, we get:

$x = \frac{-(-16) ± \sqrt{(-16)^2 - 4(1)(164)}}{2(1)}$

3. Simplifying the Expression

Now, let's simplify the expression step by step. First, we simplify the terms inside the square root and the denominator:

$x = \frac{16 ± \sqrt{256 - 656}}{2}$

Next, we calculate the value inside the square root:

$x = \frac{16 ± \sqrt{-400}}{2}$

Notice that we have a negative number inside the square root, which means the solutions will be complex numbers. This is where the imaginary unit 'i' comes into play.

4. Introducing the Imaginary Unit

Since $\sqrt{-1}$ is defined as 'i', we can rewrite $\sqrt{-400}$ as $\sqrt{400} * \sqrt{-1}$, which simplifies to $20i$. Now our equation looks like this:

$x = \frac{16 ± 20i}{2}$

5. Final Simplification and Complex Form

Finally, we can divide both terms in the numerator by 2 to get the solutions in the complex form $a + bi$:

$x = 8 ± 10i$

This gives us two solutions: $x = 8 + 10i$ and $x = 8 - 10i$. These are the complex conjugate solutions to the quadratic equation.

Expressing the Solutions in the Form $x = a + bi$ and $x = a - bi$

As we found in the step-by-step solution, the solutions to the quadratic equation $x^2 + 164 = 16x$ are complex numbers. Now, let's explicitly express these solutions in the form $x = a + bi$ and $x = a - bi$. This will help solidify our understanding of complex solutions and their conjugate pairs.

Identifying the Real and Imaginary Parts

From our previous calculations, we found the solutions to be $x = 8 + 10i$ and $x = 8 - 10i$. In the form $a + bi$, 'a' represents the real part of the complex number, and 'b' represents the imaginary part. Similarly, in the form $a - bi$, 'a' is the real part, and '-b' is the imaginary part.

For the solution $x = 8 + 10i$: The real part, a, is 8. The imaginary part, b, is 10.

For the solution $x = 8 - 10i$: The real part, a, is 8. The imaginary part, -b, is -10.

Expressing the Solutions

So, we can clearly see that the solutions are in the form $x = a + bi$ and $x = a - bi$, where $a = 8$ and $b = 10$. The solutions are complex conjugates, which is a common characteristic of quadratic equations with a negative discriminant.

Significance of Complex Conjugate Solutions

Complex conjugate solutions are crucial in many areas of mathematics and engineering. They often appear in pairs, especially when dealing with polynomials with real coefficients. Understanding and identifying these pairs is vital for solving higher-degree equations and analyzing systems in various fields. For example, in electrical engineering, complex numbers are used to represent alternating current (AC) circuits, and the conjugate solutions help in analyzing the stability and behavior of these circuits.

Importance of Understanding Complex Solutions in Mathematics

Understanding complex solutions is not just an abstract mathematical concept; it has profound implications in various fields of science and engineering. Complex numbers provide a powerful tool for solving problems that cannot be addressed using real numbers alone. Let's explore why this understanding is so important.

Applications in Various Fields

1. Electrical Engineering: In electrical engineering, complex numbers are used extensively to analyze AC circuits. The impedance, which is the opposition to the flow of current, is often represented as a complex number. Understanding complex solutions helps engineers design and analyze circuits, ensuring their stability and efficiency.
2. Quantum Mechanics: Quantum mechanics, the branch of physics dealing with the behavior of matter and energy at the atomic and subatomic levels, relies heavily on complex numbers. The wave functions that describe the state of a particle are complex-valued, and the solutions to the Schrödinger equation often involve complex numbers. These complex solutions provide insights into the probabilistic nature of quantum phenomena.
3. Fluid Dynamics: In fluid dynamics, complex potentials are used to describe two-dimensional fluid flow. Complex solutions help in analyzing the behavior of fluids around obstacles and in understanding phenomena like lift and drag.
4. Signal Processing: Signal processing, which deals with the analysis and manipulation of signals, such as audio and images, uses complex numbers for Fourier analysis. The Fourier transform, a fundamental tool in signal processing, decomposes a signal into its constituent frequencies, and these frequencies are often represented as complex numbers. Understanding complex solutions is crucial for designing filters and analyzing signal spectra.
5. Control Systems: Control systems, used in robotics, automation, and aerospace engineering, often involve complex transfer functions. Complex solutions help in analyzing the stability and performance of these systems. By understanding the behavior of the system in the complex plane, engineers can design controllers that ensure the system operates as desired.

The Role of the Discriminant

The discriminant ($b^2 - 4ac$) plays a crucial role in determining the nature of the solutions to a quadratic equation. As we discussed earlier, if the discriminant is negative, the solutions are complex. This is because the square root of a negative number is an imaginary number. The discriminant not only tells us whether the solutions are complex but also provides insight into the behavior of the quadratic function. If $b^2 - 4ac > 0$, there are two distinct real roots. If $b^2 - 4ac = 0$, there is exactly one real root (a repeated root). If $b^2 - 4ac < 0$, there are two complex conjugate roots.

Building a Stronger Mathematical Foundation

By mastering the concept of complex solutions, you build a stronger foundation for more advanced mathematical topics. Complex analysis, a branch of mathematics that deals with complex functions, is essential in many areas of physics and engineering. A solid understanding of complex numbers and their solutions is a prerequisite for delving into these advanced topics. Moreover, understanding complex solutions enhances your problem-solving skills and broadens your mathematical toolkit. It allows you to approach a wider range of problems with confidence and creativity.

Conclusion

In summary, we've successfully solved the quadratic equation $x^2 + 164 = 16x$ and expressed the solutions in the complex form $x = a + bi$ and $x = a - bi$. The solutions are $x = 8 + 10i$ and $x = 8 - 10i$. We accomplished this by rearranging the equation into standard form, applying the quadratic formula, simplifying the expression, and introducing the imaginary unit 'i'. We also discussed the importance of understanding complex solutions in various fields of mathematics, science, and engineering. Mastering these concepts is crucial for anyone looking to deepen their understanding of mathematics and its applications.

For further exploration of quadratic equations and complex numbers, you can visit Khan Academy's resource on Quadratic Equations.