Solving Quadratic Equations: Finding Complex Solutions

by Alex Johnson 55 views

Introduction

In the realm of mathematics, quadratic equations hold a significant position, appearing in various contexts ranging from simple algebra to advanced calculus. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation, also known as roots or zeros, represent the values of x that satisfy the equation. These solutions can be real numbers or complex numbers, depending on the discriminant of the equation. Understanding how to find complex solutions is crucial for a comprehensive understanding of quadratic equations and their applications.

This article delves into the process of finding all complex solutions of a quadratic equation, using the specific example of x² - 5x + 7 = 0. We will explore the quadratic formula, a powerful tool for solving quadratic equations, and demonstrate its application in obtaining both real and complex solutions. By the end of this discussion, you will gain a solid understanding of how to tackle quadratic equations and extract their complex roots.

Understanding the Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations of the form ax² + bx + c = 0. This formula provides a direct method to find the roots of the equation, regardless of whether they are real or complex. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

Where:

  • x represents the solutions or roots of the quadratic equation.
  • a, b, and c are the coefficients of the quadratic equation.
  • The symbol '±' indicates that there are two possible solutions, one with addition and one with subtraction.
  • The expression b² - 4ac is known as the discriminant, which plays a crucial role in determining the nature of the solutions.

The discriminant, b² - 4ac, provides valuable information about the roots of the quadratic equation:

  • If b² - 4ac > 0, the equation has two distinct real solutions.
  • If b² - 4ac = 0, the equation has one real solution (a repeated root).
  • If b² - 4ac < 0, the equation has two complex solutions.

The presence of a negative value under the square root in the quadratic formula indicates that the solutions will be complex numbers. Complex numbers involve the imaginary unit, denoted by i, where i² = -1. Understanding the quadratic formula and the role of the discriminant is essential for solving quadratic equations and identifying the nature of their solutions.

Applying the Quadratic Formula to x² - 5x + 7 = 0

To find the complex solutions of the equation x² - 5x + 7 = 0, we will apply the quadratic formula. First, we need to identify the coefficients a, b, and c in the equation.

In this case:

  • a = 1 (the coefficient of x²)
  • b = -5 (the coefficient of x)
  • c = 7 (the constant term)

Now, we substitute these values into the quadratic formula:

x = (-(-5) ± √((-5)² - 4 * 1 * 7)) / (2 * 1)

Simplifying the expression:

x = (5 ± √(25 - 28)) / 2

x = (5 ± √(-3)) / 2

Since we have a negative value under the square root, we know that the solutions will be complex. We can express the square root of -3 using the imaginary unit i:

√(-3) = √(-1 * 3) = √(-1) * √(3) = i√(3)

Substituting this back into the equation:

x = (5 ± i√(3)) / 2

Therefore, the two complex solutions are:

  • x₁ = (5 + i√(3)) / 2
  • x₂ = (5 - i√(3)) / 2

These solutions are complex conjugates of each other, which is a common characteristic of complex roots in quadratic equations with real coefficients. By applying the quadratic formula and understanding the properties of complex numbers, we have successfully found the complex solutions to the equation x² - 5x + 7 = 0.

Verifying the Complex Solutions

To ensure the accuracy of our solutions, we can verify them by substituting them back into the original equation, x² - 5x + 7 = 0. Let's verify the first solution, x₁ = (5 + i√(3)) / 2:

Substituting x₁ into the equation:

((5 + i√(3)) / 2)² - 5 * ((5 + i√(3)) / 2) + 7 = 0

Expanding the terms:

((25 + 10i√(3) - 3) / 4) - (25 + 5i√(3)) / 2 + 7 = 0

Simplifying:

(22 + 10i√(3)) / 4 - (25 + 5i√(3)) / 2 + 7 = 0

Multiplying through by 4 to eliminate fractions:

22 + 10i√(3) - 2(25 + 5i√(3)) + 28 = 0

22 + 10i√(3) - 50 - 10i√(3) + 28 = 0

Combining like terms:

(22 - 50 + 28) + (10i√(3) - 10i√(3)) = 0

0 = 0

The equation holds true, confirming that x₁ = (5 + i√(3)) / 2 is indeed a solution. A similar process can be followed to verify the second solution, x₂ = (5 - i√(3)) / 2, which will also satisfy the equation. This verification step is crucial in ensuring the correctness of the obtained solutions and reinforcing the understanding of how complex numbers interact within quadratic equations.

Nature of Complex Solutions

When a quadratic equation has complex solutions, these solutions always come in conjugate pairs. This means that if a + bi is a solution, then a - bi is also a solution, where a and b are real numbers and i is the imaginary unit. In our example, the solutions (5 + i√(3)) / 2 and (5 - i√(3)) / 2 are complex conjugates.

Complex solutions arise when the discriminant (b² - 4ac) of the quadratic equation is negative. A negative discriminant indicates that the square root in the quadratic formula will involve the imaginary unit i, leading to complex roots. Understanding the nature of complex solutions is crucial for interpreting the results obtained from the quadratic formula and recognizing the inherent symmetry in the solutions.

Complex solutions have significant applications in various fields, including electrical engineering, quantum mechanics, and signal processing. In these areas, complex numbers are used to model oscillatory behavior, wave phenomena, and other complex systems. The ability to find and interpret complex solutions of quadratic equations is therefore a valuable skill in many scientific and engineering disciplines.

Conclusion

In this article, we have explored the process of finding complex solutions to quadratic equations, using the example of x² - 5x + 7 = 0. We began by introducing the quadratic formula, a fundamental tool for solving quadratic equations, and highlighting the significance of the discriminant in determining the nature of the solutions. By applying the quadratic formula to our example equation, we obtained two complex solutions: (5 + i√(3)) / 2 and (5 - i√(3)) / 2.

We then verified these solutions by substituting them back into the original equation, confirming their validity. Furthermore, we discussed the nature of complex solutions, emphasizing that they always come in conjugate pairs and arise when the discriminant is negative. Understanding complex solutions is essential for a comprehensive understanding of quadratic equations and their applications in various fields.

The ability to solve quadratic equations and extract both real and complex roots is a valuable skill in mathematics and its applications. The quadratic formula provides a robust and reliable method for finding these solutions, and a thorough understanding of the discriminant allows us to predict the nature of the roots. By mastering these concepts, you will be well-equipped to tackle a wide range of mathematical problems involving quadratic equations.

For further exploration of quadratic equations and complex numbers, consider visiting Khan Academy's Algebra II section. This resource provides comprehensive lessons, practice exercises, and videos to deepen your understanding of these important mathematical concepts.