Solving Quadratic Equations With Complex Roots Step-by-Step

Quadratic equations are a fundamental concept in algebra, and understanding how to solve them is crucial for various mathematical and scientific applications. While some quadratic equations have real number solutions, others have complex number solutions, often involving the imaginary unit i, where i² = -1. In this article, we will delve into how to solve quadratic equations with complex roots, using the quadratic formula and illustrating the process with a step-by-step example. Understanding complex roots enhances one’s problem-solving capabilities and deepens comprehension of mathematical intricacies. This exploration will cover everything from the underlying theory to practical application.

The Quadratic Formula: A Key to Unlocking Solutions

The quadratic formula is a cornerstone in algebra for solving equations of the form ax² + bx + c = 0. This formula is particularly valuable when dealing with equations that cannot be easily factored. Let’s understand its utility and application thoroughly.

Derivation and Significance

The quadratic formula is derived by completing the square on the general form of the quadratic equation, ax² + bx + c = 0. This method rearranges the equation into a perfect square, making it solvable for x. The resulting formula is:

x = (-b ± √(b² - 4ac)) / (2a)

The quadratic formula is significant because it provides a straightforward method to find the roots of any quadratic equation, regardless of whether the roots are real or complex. This formula simplifies the process of finding solutions and is essential for more advanced mathematical topics.

Applying the Formula

To apply the quadratic formula, you must first identify the coefficients a, b, and c from the quadratic equation. Once identified, these values are plugged into the formula, and the equation is simplified to find the solutions for x. The discriminant, (b² - 4ac), plays a crucial role in determining the nature of the roots. If the discriminant is negative, the roots will be complex, indicating the presence of imaginary numbers in the solution. Let’s explore how this works with an example to clarify the process.

The Discriminant: Unveiling the Nature of Roots

The discriminant (b² - 4ac) is a critical component within the quadratic formula that determines the nature of the roots of a quadratic equation. The value of the discriminant reveals whether the roots are real, distinct, repeated, or complex. This understanding is crucial for accurately interpreting the solutions of quadratic equations.

Understanding the Discriminant’s Role

The discriminant helps us classify the roots as follows:

• If b² - 4ac > 0: The equation has two distinct real roots.
• If b² - 4ac = 0: The equation has exactly one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex roots, which are complex conjugates of each other.

The sign of the discriminant provides immediate insights into the type of solutions to expect, making it a valuable tool in problem-solving. Understanding this can save time and effort by guiding the solution process appropriately.

Practical Implications

When solving a quadratic equation, calculating the discriminant first can help anticipate the nature of the roots. This anticipation allows for the application of suitable methods for finding the roots. For instance, if the discriminant is negative, we know we will be working with complex numbers and can adjust our calculations accordingly. This preliminary step enhances both the accuracy and efficiency of solving quadratic equations.

Complex Roots: Stepping into the Imaginary Realm

When the discriminant (b² - 4ac) is negative, the quadratic equation has complex roots. Complex roots involve the imaginary unit i, where i² = -1. Understanding and working with complex numbers is essential for solving such equations.

Defining Complex Numbers

A complex number is expressed in the form a + bi, where a is the real part and bi is the imaginary part. The imaginary unit i allows us to express the square root of negative numbers, which is crucial for solving equations with a negative discriminant. Complex numbers extend the number system beyond real numbers, providing solutions for equations that would otherwise be unsolvable in the real number system.

Handling Negative Square Roots

To handle a negative square root, such as √(-n), we express it as √(n) i. For example, √(-16) = √16 i = 4i. This transformation allows us to include imaginary numbers in our calculations and express solutions that involve the square root of negative numbers. Mastery of this technique is fundamental for dealing with complex roots effectively.

Complex Conjugates

Complex roots of quadratic equations with real coefficients always occur in conjugate pairs. If a + bi is a root, then its complex conjugate a - bi is also a root. This property stems from the quadratic formula, where the ± sign before the square root leads to two solutions that differ only in the sign of the imaginary part. Understanding complex conjugates simplifies the process of identifying all roots of a quadratic equation once one complex root is found.

Step-by-Step Solution: Finding the Complex Roots

Let’s solve the given equation, 2x² - 4x + 9 = 0, step-by-step to illustrate how to find complex roots using the quadratic formula.

1. Identify Coefficients

First, we identify the coefficients a, b, and c from the equation 2x² - 4x + 9 = 0:

• a = 2
• b = -4
• c = 9

Identifying these coefficients correctly is crucial for accurate substitution into the quadratic formula.

2. Apply the Quadratic Formula

Next, we substitute the values of a, b, and c into the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

x = (-(-4) ± √((-4)² - 4(2)(9))) / (2(2))

This substitution sets up the equation for simplification and solution.

3. Simplify the Expression

Now, we simplify the expression step-by-step:

x = (4 ± √(16 - 72)) / 4

x = (4 ± √(-56)) / 4

Simplifying the discriminant under the square root is a key step in determining the nature of the roots.

4. Handle the Negative Square Root

Since we have a negative number under the square root, we express it using the imaginary unit i:

x = (4 ± √(56) i) / 4

x = (4 ± √(4 * 14) i) / 4

x = (4 ± 2√(14) i) / 4

Converting the negative square root into an imaginary number allows us to proceed with finding the complex roots.

5. Simplify to Final Form

Finally, we simplify the expression to its simplest form:

x = (4 / 4) ± (2√(14) i / 4)

x = 1 ± (√(14) / 2) i

Thus, the roots are 1 + (√(14) / 2) i and 1 - (√(14) / 2) i.

6. Identify the Correct Answer

Comparing our solutions with the given options, we find that 1 + (√(14) / 2) i matches option D.

Therefore, the correct answer is:

D. 1 + (√(14) / 2) i

Conclusion

Solving quadratic equations with complex roots involves understanding and applying the quadratic formula, recognizing the role of the discriminant, and correctly handling imaginary numbers. By following a systematic approach, one can efficiently find both real and complex solutions. Mastering these techniques not only enhances mathematical proficiency but also provides a deeper understanding of algebraic principles. The process outlined here provides a comprehensive guide to tackling such problems, ensuring accuracy and efficiency. The ability to solve quadratic equations, especially those with complex roots, is a fundamental skill in mathematics and has applications in various fields, including physics and engineering. Therefore, a thorough understanding and practice of these concepts are invaluable for anyone pursuing studies or careers in these areas. The journey through quadratic equations, complete with real and complex roots, highlights the richness and versatility of algebra as a problem-solving tool.

Practice Question

What is one root of this equation? 2 x² - 4 x + 9 = 0 A. -2 + (√(14) / 2) i B. -1 + (√(14) / 2) i C. 1 + (√(14) / 3) i D. 1 + (√(14) / 2) i

Answer: D. 1 + (√(14) / 2) i