Solving Z^2 = 7i Expressing The Answer In Cartesian Form

In the realm of complex numbers, solving equations often requires a blend of algebraic manipulation and a deep understanding of the properties of imaginary units. One such intriguing equation is z^2 = 7i, where z is a complex number and i is the imaginary unit (√-1). This article delves into the step-by-step process of solving this equation, expressing the solution in the Cartesian form (a + bi), where 'a' and 'b' are real numbers. Understanding the nuances of complex number arithmetic and the representation of complex numbers in the complex plane is crucial for tackling such problems. We will explore how to leverage these concepts to find the roots of the given equation.

Understanding Complex Numbers and the Cartesian Form

To effectively solve the equation z^2 = 7i, it's essential to grasp the fundamentals of complex numbers. A complex number is expressed in the form a + bi, where 'a' represents the real part and 'b' represents the imaginary part. The imaginary unit, denoted by i, is defined as the square root of -1 (i = √-1), which implies that i² = -1. This unique property of i allows us to extend the number system beyond real numbers and venture into the complex plane.

The Cartesian form, also known as the rectangular form, is a way of representing complex numbers using the real and imaginary parts as coordinates in a two-dimensional plane called the complex plane. The horizontal axis represents the real part ('a'), and the vertical axis represents the imaginary part ('b'). Thus, a complex number a + bi can be visualized as a point (a, b) in the complex plane. This geometric interpretation is immensely helpful in understanding the behavior of complex numbers and their operations.

When dealing with equations involving complex numbers, the Cartesian form provides a convenient framework for algebraic manipulation. By expressing complex numbers in this form, we can apply familiar algebraic rules and techniques, such as equating real and imaginary parts, to solve for unknown variables. The equation z^2 = 7i serves as an excellent example of how the Cartesian form can be utilized to find the complex roots of an equation.

Setting up the Solution: Expressing z in Cartesian Form

Our initial step towards solving z^2 = 7i involves expressing the complex number z in its Cartesian form. Let's assume that z can be written as z = x + yi, where x and y are real numbers. This representation is the cornerstone of our approach, allowing us to translate the complex equation into a system of real equations that we can solve using standard algebraic methods.

By substituting z = x + yi into the original equation, we obtain (x + yi)² = 7i. This substitution transforms the problem into one involving the squares and products of real and imaginary terms. The next step is to expand the left-hand side of the equation, which will reveal how the real and imaginary parts interact and how we can separate them to form a system of equations.

This step is crucial because it allows us to break down a single complex equation into two real equations, which are often easier to handle. By equating the real and imaginary parts on both sides of the equation, we can establish a relationship between x and y that will ultimately lead us to the solutions for z. The careful expansion and separation of terms are key to navigating the complexities of this problem.

Expanding and Equating Real and Imaginary Parts

With z expressed in Cartesian form as z = x + yi, the equation z^2 = 7i becomes (x + yi)^2 = 7i. To proceed, we expand the left-hand side using the binomial formula or simply by multiplying the expression by itself: (x + yi)(x + yi) = x^2 + 2xyi + (yi)^2. Since i² = -1, this simplifies to x^2 + 2xyi - y^2. Now, we have the equation x^2 + 2xyi - y^2 = 7i.

The next crucial step is to equate the real and imaginary parts on both sides of the equation. On the left-hand side, the real part is x^2 - y^2, and the imaginary part is 2xy. On the right-hand side, the real part is 0 (since there is no real term), and the imaginary part is 7. This leads us to two equations:

1. x^2 - y^2 = 0 (Equating the real parts)
2. 2xy = 7 (Equating the imaginary parts)

These two equations form a system of non-linear equations that we need to solve for x and y. The first equation tells us that x^2 = y^2, which implies that x = y or x = -y. The second equation provides a relationship between x and y through their product. Solving this system will give us the values of x and y that satisfy the original equation.

Solving the System of Equations

We now have the system of equations:

1. x^2 - y^2 = 0
2. 2xy = 7

From the first equation, x^2 = y^2, we deduce two possibilities: x = y or x = -y. Let's consider each case separately.

Case 1: x = y

Substituting x = y into the second equation, 2xy = 7, we get 2x^2 = 7. Solving for x, we find x^2 = 7/2, so x = ±√(7/2). Since x = y, we have two solutions in this case: (x, y) = (√(7/2), √(7/2)) and (x, y) = (-√(7/2), -√(7/2)).

Case 2: x = -y

Substituting x = -y into the second equation, 2xy = 7, we get -2y^2 = 7. This implies y^2 = -7/2, which has no real solutions for y because the square of a real number cannot be negative. Therefore, this case does not yield any valid solutions.

Thus, we have found two pairs of real numbers (x, y) that satisfy the system of equations. These pairs correspond to the complex numbers that are the solutions to the original equation z^2 = 7i. The next step is to express these solutions in the standard Cartesian form.

Expressing the Solutions in Cartesian Form

From the previous step, we found two solutions for the system of equations derived from z^2 = 7i: (x, y) = (√(7/2), √(7/2)) and (x, y) = (-√(7/2), -√(7/2)). Recall that we expressed z in the Cartesian form as z = x + yi. Now, we can substitute the values of x and y to obtain the solutions for z.

For the first solution, (x, y) = (√(7/2), √(7/2)), we have:

z₁ = √(7/2) + √(7/2)i

For the second solution, (x, y) = (-√(7/2), -√(7/2)), we have:

z₂ = -√(7/2) - √(7/2)i

These are the two complex numbers that satisfy the equation z^2 = 7i, expressed in the Cartesian form. We can further simplify these expressions by rationalizing the denominators or by factoring out common terms, but the current form clearly shows the real and imaginary parts of each solution.

The solutions z₁ and z₂ are complex conjugates of each other, which is a common occurrence when solving quadratic equations with complex roots. This relationship can be verified by observing that the real parts are the same, while the imaginary parts have opposite signs. The fact that we obtained two solutions is consistent with the fundamental theorem of algebra, which states that a polynomial equation of degree n has n complex roots (counting multiplicities).

Verification of the Solutions

To ensure the accuracy of our solutions, it is essential to verify that z₁ = √(7/2) + √(7/2)i and z₂ = -√(7/2) - √(7/2)i indeed satisfy the original equation, z^2 = 7i. We can do this by squaring each solution and checking if the result is equal to 7i.

Let's start with z₁ = √(7/2) + √(7/2)i:

z₁^2 = (√(7/2) + √(7/2)i)^2

Expanding this, we get:

z₁^2 = (√(7/2))^2 + 2(√(7/2))(√(7/2)i) + (√(7/2)i)^2

z₁^2 = 7/2 + 2(7/2)i + (7/2)i^2

Since i² = -1, we have:

z₁^2 = 7/2 + 7i - 7/2 = 7i

Thus, z₁ is indeed a solution.

Now, let's verify z₂ = -√(7/2) - √(7/2)i:

z₂^2 = (-√(7/2) - √(7/2)i)^2

Expanding this, we get:

z₂^2 = (-√(7/2))^2 + 2(-√(7/2))(-√(7/2)i) + (-√(7/2)i)^2

z₂^2 = 7/2 + 2(7/2)i + (7/2)i^2

Again, since i² = -1, we have:

z₂^2 = 7/2 + 7i - 7/2 = 7i

Thus, z₂ is also a solution. This verification confirms that our solutions are correct and that we have successfully solved the equation z^2 = 7i.

Conclusion

In conclusion, solving the equation z^2 = 7i in Cartesian form involves a series of steps that highlight the interplay between algebra and complex number theory. By expressing z as x + yi, we transformed the complex equation into a system of real equations. Solving this system yielded two solutions: z₁ = √(7/2) + √(7/2)i and z₂ = -√(7/2) - √(7/2)i. These solutions, when squared, result in 7i, thereby confirming their validity.

This problem exemplifies the power of the Cartesian form in handling complex number equations. It also underscores the importance of understanding the properties of the imaginary unit i and the geometric interpretation of complex numbers in the complex plane. The techniques employed in this solution can be applied to a wide range of complex number problems, making it a valuable tool in mathematics and related fields.

The process of solving z^2 = 7i not only provides a concrete answer but also enhances our understanding of complex number arithmetic and the methods for tackling such problems. The ability to manipulate complex numbers and express them in various forms is a fundamental skill in many areas of mathematics, physics, and engineering. This article has demonstrated a systematic approach to solving a complex number equation, providing a solid foundation for further exploration in this fascinating area of mathematics.