Solving X² - 8x + 32 = 0: Find Roots In A + Bi Form
Understanding how to solve quadratic equations is a fundamental skill in algebra. Quadratic equations, characterized by the general form ax² + bx + c = 0, often appear in various mathematical and real-world contexts. In this comprehensive guide, we'll break down the process of finding the roots of the specific equation x² - 8x + 32 = 0 and express them in the simplest a + bi form. This involves a step-by-step approach, utilizing the quadratic formula, and simplifying the result to achieve the desired format. Let’s dive deep into the methods and concepts required to master this type of problem.
Understanding the Quadratic Formula
The quadratic formula is a cornerstone for solving quadratic equations. It provides a direct method to find the roots, regardless of whether they are real or complex. This formula is particularly useful when factoring is not straightforward or when dealing with equations that have complex roots. The formula is derived from the process of completing the square and is universally applicable to all quadratic equations. Before we delve into solving our specific equation, let’s first understand the formula and its components in detail.
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
Where:
- a, b, and c are the coefficients from the quadratic equation ax² + bx + c = 0.
- The symbol '±' indicates that there are two possible solutions, one with addition and one with subtraction.
- The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant determines the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has one real root (a repeated root).
- If b² - 4ac < 0, the equation has two complex roots.
In our case, the equation is x² - 8x + 32 = 0, so we can identify the coefficients as follows:
- a = 1
- b = -8
- c = 32
Understanding these coefficients is the first step in applying the quadratic formula. We will now substitute these values into the formula to find the roots of the equation. This process will involve careful calculation and simplification, particularly when dealing with the square root of a negative number, which indicates complex roots. The quadratic formula is not just a tool for finding solutions; it is a powerful concept that underpins much of algebraic problem-solving. By mastering its application, students gain a deeper understanding of quadratic equations and their properties. Remember, practice is key, so working through various examples will help solidify your understanding and improve your problem-solving skills.
Applying the Quadratic Formula to x² - 8x + 32 = 0
Now that we have a solid grasp of the quadratic formula, let's apply it to our specific equation: x² - 8x + 32 = 0. We have already identified the coefficients:
- a = 1
- b = -8
- c = 32
Substituting these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)² - 4 * 1 * 32)) / (2 * 1)
Let's break this down step by step:
- Simplify the terms inside the parentheses: x = (8 ± √(64 - 128)) / 2
- Calculate the discriminant (the expression inside the square root): x = (8 ± √(-64)) / 2
Notice that the discriminant is negative (-64). This indicates that the roots of the equation are complex numbers. Complex numbers involve the imaginary unit, denoted as i, where i is defined as the square root of -1 (i = √-1). Dealing with complex numbers requires understanding their properties and how they interact with mathematical operations. In this case, the negative discriminant tells us we will have roots that include an imaginary component. This is a crucial piece of information because it guides us in how to simplify the expression further.
- Express the square root of the negative number in terms of i: √(-64) = √(64 * -1) = √64 * √-1 = 8i
- Substitute this back into the equation: x = (8 ± 8i) / 2
Now, we have an expression that involves both real and imaginary parts. The next step is to simplify this expression to get the roots in the simplest a + bi form. This involves dividing both the real and imaginary parts by the denominator. Simplifying complex expressions is a crucial skill in algebra, particularly when dealing with quadratic equations that have non-real roots. The ability to manipulate and simplify these expressions accurately is essential for solving a wide range of mathematical problems. The quadratic formula provides a robust method for finding roots, but understanding how to handle complex numbers is key to expressing those roots in their simplest form.
Simplifying to the a + bi Form
We've arrived at the expression x = (8 ± 8i) / 2. To express the roots in the simplest a + bi form, we need to divide both the real and imaginary parts by the denominator. This step is crucial for presenting the solution in a standard and easily understandable format. The a + bi form is a conventional way to represent complex numbers, where a represents the real part and bi represents the imaginary part. This form allows for easy comparison and manipulation of complex numbers in various mathematical contexts.
Let's perform the division:
x = 8/2 ± (8i)/2
Simplify each term:
x = 4 ± 4i
This gives us two complex roots:
- x₁ = 4 + 4i
- x₂ = 4 - 4i
These are the roots of the equation x² - 8x + 32 = 0, expressed in the simplest a + bi form. Notice that the roots are complex conjugates of each other. Complex conjugates are pairs of complex numbers with the same real part but opposite imaginary parts. This is a common occurrence when solving quadratic equations with real coefficients and a negative discriminant. Identifying complex conjugate pairs can often serve as a check for the correctness of your solution. Presenting the solutions in this clear and concise format not only answers the question but also demonstrates a thorough understanding of complex numbers and their representation.
Verifying the Roots
To ensure the accuracy of our solutions, we can verify them by substituting each root back into the original equation. This step is an excellent practice for reinforcing your understanding and minimizing errors. Substituting complex roots into a quadratic equation can seem daunting, but it's a valuable skill that enhances your algebraic proficiency. Let’s start by verifying the first root, x₁ = 4 + 4i.
Substitute x₁ = 4 + 4i into x² - 8x + 32 = 0:
(4 + 4i)² - 8(4 + 4i) + 32 = 0
Expand the terms:
(16 + 32i + 16i²) - 32 - 32i + 32 = 0
Remember that i² = -1, so replace 16i² with -16:
(16 + 32i - 16) - 32 - 32i + 32 = 0
Simplify the equation:
16 + 32i - 16 - 32 - 32i + 32 = 0
Combine like terms:
(16 - 16 - 32 + 32) + (32i - 32i) = 0
0 + 0i = 0
0 = 0
Since the equation holds true, x₁ = 4 + 4i is indeed a root of the equation. Now, let's verify the second root, x₂ = 4 - 4i, using the same process. This will further validate our solution and provide additional confidence in our calculations. Consistent verification of solutions is a hallmark of careful mathematical practice, leading to more accurate and reliable results. By taking the time to verify our roots, we not only confirm the correctness of our solution but also deepen our understanding of the algebraic principles involved.
Substitute x₂ = 4 - 4i into x² - 8x + 32 = 0:
(4 - 4i)² - 8(4 - 4i) + 32 = 0
Expand the terms:
(16 - 32i + 16i²) - 32 + 32i + 32 = 0
Replace 16i² with -16 (since i² = -1):
(16 - 32i - 16) - 32 + 32i + 32 = 0
Simplify the equation:
16 - 32i - 16 - 32 + 32i + 32 = 0
Combine like terms:
(16 - 16 - 32 + 32) + (-32i + 32i) = 0
0 + 0i = 0
0 = 0
The equation holds true for x₂ = 4 - 4i as well, confirming that it is also a root of the given quadratic equation. This verification process not only assures us of the correctness of our solutions but also highlights the importance of thoroughness in mathematical problem-solving. By consistently checking our work, we minimize the likelihood of errors and build a stronger foundation in algebraic techniques. The ability to confidently verify solutions is a key skill for any mathematics student, promoting both accuracy and a deeper understanding of the underlying concepts.
Conclusion
In summary, we've successfully found the roots of the quadratic equation x² - 8x + 32 = 0 and expressed them in the simplest a + bi form. By understanding and applying the quadratic formula, we navigated through the complexities of complex numbers and derived the roots 4 + 4i and 4 - 4i. We also emphasized the importance of verifying solutions to ensure accuracy and deepen understanding. This process not only provides the correct answer but also reinforces fundamental algebraic principles.
Mastering the quadratic formula and working with complex numbers are crucial skills in mathematics. They form the foundation for more advanced topics and are frequently encountered in various scientific and engineering applications. The ability to confidently solve quadratic equations, whether they have real or complex roots, is a testament to a strong mathematical foundation. Remember, practice is essential. The more you work with these concepts, the more proficient you will become.
Quadratic equations and complex numbers are not just abstract concepts; they have real-world applications in fields such as physics, engineering, and computer science. Understanding these concepts allows you to model and solve problems in a wide range of contexts. Whether you are calculating projectile motion, designing electrical circuits, or developing algorithms, the principles you've learned here will prove invaluable.
As you continue your mathematical journey, remember that each problem solved and each concept understood builds a stronger foundation for future learning. Don't be discouraged by challenges; instead, view them as opportunities to grow and deepen your knowledge. With consistent effort and a willingness to explore, you will unlock the power of mathematics and its ability to explain and shape the world around us.
For further exploration and resources on quadratic equations and complex numbers, consider visiting Khan Academy's Algebra I section, which offers comprehensive lessons and practice exercises.
