Solve for [tex]$z^2=7 i$[/tex] expressing your answer in the Cartesian form
Answer (1)
Assume z = x + y i and substitute into z 2 = 7 i .
Expand and equate real and imaginary parts: x 2 − y 2 = 0 and 2 x y = 7 .
Solve the system of equations to find x = y = ± 2 7 .
The solutions are z = 2 7 + i 2 7 and z = − 2 7 − i 2 7 . The final answer is 2 7 + i 2 7 , − 2 7 − i 2 7 .
Explanation
Understanding the Problem We are given the equation z 2 = 7 i , where z is a complex number. Our goal is to find the solution for z in the Cartesian form, which means we want to express z as z = x + y i , where x and y are real numbers.
Substituting z = x + yi Let's assume that z = x + y i , where x and y are real numbers. We substitute this into the given equation z 2 = 7 i . This gives us ( x + y i ) 2 = 7 i .
Expanding the Square Now, we expand the left side of the equation: ( x + y i ) 2 = x 2 + 2 x y i + ( y i ) 2 = x 2 + 2 x y i − y 2 = ( x 2 − y 2 ) + 2 x y i .
Equating Real and Imaginary Parts We now have the equation ( x 2 − y 2 ) + 2 x y i = 7 i . For two complex numbers to be equal, their real and imaginary parts must be equal. Therefore, we can equate the real and imaginary parts of the equation:
Real part: x 2 − y 2 = 0 Imaginary part: 2 x y = 7
Solving for x and y From the first equation, x 2 − y 2 = 0 , we have x 2 = y 2 . This implies that x = y or x = − y .
Let's consider the case where x = y . Substituting this into the second equation, 2 x y = 7 , we get 2 x 2 = 7 . Solving for x , we have x 2 = 2 7 , so x = ± 2 7 . Since x = y , we have two solutions: x = y = 2 7 or x = y = − 2 7 .
Now, let's consider the case where x = − y . Substituting this into the second equation, 2 x y = 7 , we get 2 x ( − x ) = 7 , which simplifies to − 2 x 2 = 7 . This gives x 2 = − 2 7 . Since x is a real number, x 2 cannot be negative. Therefore, there are no real solutions for x in this case.
Finding the Solutions From the previous step, we found two solutions for z :
x = y = 2 7 , which gives z = 2 7 + i 2 7
x = y = − 2 7 , which gives z = − 2 7 − i 2 7
We can approximate 2 7 = 3.5 ≈ 1.8708 . Therefore, the solutions are approximately z = 1.8708 + 1.8708 i and z = − 1.8708 − 1.8708 i .
Final Answer Therefore, the solutions for z 2 = 7 i in Cartesian form are:
z = 2 7 + i 2 7 and z = − 2 7 − i 2 7 .
Examples
Complex numbers are used extensively in electrical engineering to analyze alternating current (AC) circuits. Impedance, which is the opposition to the flow of current in an AC circuit, is represented as a complex number. Solving equations involving complex numbers, like the one in this problem, helps engineers determine the voltage and current relationships in AC circuits, ensuring efficient and safe operation of electrical devices and power systems. For example, if you are designing a circuit and need a specific current flow, you would use complex numbers to calculate the required impedance.
